Introduction to mechanics_and_

1. This is page i Printer: Opaque this Introduction to Mechanics and Symmetry A Basic Exposition of Classical Mechanical Systems Second Edition Jerrold E. Marsden and Tudor S. Ratiu Last modiﬁed on 15 July 1998

2. v To Barbara and Lilian for their love and support ........................... 15 July 1998—18h02 ...........................

3. vi ........................... 15 July 1998—18h02 ...........................

4. This is page ix Printer: Opaque this Preface Symmetry and mechanics have been close partners since the time of the founding masters: Newton, Euler, Lagrange, Laplace, Poisson, Jacobi, Ha- milton, Kelvin, Routh, Riemann, Noether, Poincar´, Einstein, Schr¨dinger, e o Cartan, Dirac, and to this day, symmetry has continued to play a strong role, especially with the modern work of Kolmogorov, Arnold, Moser, Kir- illov, Kostant, Smale, Souriau, Guillemin, Sternberg, and many others. This book is about these developments, with an emphasis on concrete applications that we hope will make it accessible to a wide variety of readers, especially senior undergraduate and graduate students in science and engineering. The geometric point of view in mechanics combined with solid analysis has been a phenomenal success in linking various diverse areas, both within and across standard disciplinary lines. It has provided both insight into fundamental issues in mechanics (such as variational and Hamiltonian structures in continuum mechanics, fluid mechanics, and plasma physics) and provided useful tools in specific models such as new stability and bifurcation criteria using the energy-Casimir and energy-momentum methods, new numerical codes based on geometrically exact update procedures and variational integrators, and new reorientation techniques in control theory and robotics. Symmetry was already widely used in mechanics by the founders of the subject, and has been developed considerably in recent times in such diverse phenomena as reduction, stability, bifurcation and solution symmetry breaking relative to a given system symmetry group, methods of finding explicit solutions for integrable systems, and a deeper understanding of spe-

5. x Preface cial systems, such as the Kowalewski top. We hope this book will provide a reasonable avenue to, and foundation for, these exciting developments. Because of the extensive and complex set of possible directions in which one can develop the theory, we have provided a fairly lengthy introduction. It is intended to be read lightly at the beginning and then consulted from time to time as the text itself is read. This volume contains much of the basic theory of mechanics and should prove to be a useful foundation for further, as well as more specialized topics. Due to space limitations we warn the reader that many important topics in mechanics are not treated in this volume. We are preparing a second volume on general reduction theory and its applications. With luck, a little support, and yet more hard work, it will be available in the near future. Solutions Manual. A solution manual is available for insturctors that contains complete solutions to many of the exercises and other supplemen- tary comments. This may be obtained from the publisher. Internet Supplements. To keep the size of the book within reason, we are making some material available (free) on the internet. These are a collection of sections whose omission does not interfere with the main flow of the text. See http://www.cds.caltech.edu/~marsden. Updates and information about the book can also be found there. What is New in the Second Edition? In this second edition, the main structural changes are the creation of the Solutions manual (along with many more Exercises in the text) and the internet supplements. The internet supplements contain, for example, the material on the Maslov index that was not needed for the main flow of the book. As for the substance of the text, much of the book was rewritten throughout to improve the flow of material and to correct inaccuracies. Some examples: the material on the Hamilton-Jacobi theory was completely rewritten, a new section on Routh reduction (§8.9) was added, Chapter 9 on Lie groups was substantially improved and expanded and the presentation of examples of coadjoint orbits (Chapter 14) was improved by stressing matrix methods throughout. Acknowledgments. We thank Alan Weinstein, Rudolf Schmid, and Rich Spencer for helping with an early set of notes that helped us on our way. Our many colleagues, students, and readers, especially Henry Abarbanel, Vladimir Arnold, Larry Bates, Michael Berry, Tony Bloch, Hans Duister- maat, Marty Golubitsky, Mark Gotay, George Haller, Aaron Hershman, Darryl Holm, Phil Holmes, Sameer Jalnapurkar, Edgar Knobloch, P.S. Krishnaprasad, Naomi Leonard, Debra Lewis, Robert Littlejohn, Richard Montgomery, Phil Morrison, Richard Murray, Peter Olver, Oliver O’Reilly, Juan-Pablo Ortega, George Patrick, Octavian Popp, Matthias Reinsch, Shankar Sastry, Juan Simo, Hans Troger, and Steve Wiggins have our deep- est gratitude for their encouragement and suggestions. We also collectively ........................... 15 July 1998—18h02 ...........................

6. xi thank all our students and colleagues who have used these notes and have provided valuable advice. We are also indebted to Carol Cook, Anne Kao, Nawoyuki Gregory Kubota, Sue Knapp, Barbara Marsden, Marnie McEl- hiney, June Meyermann, Teresa Wild, and Ester Zack for their dedicated and patient work on the typesetting and artwork for this book. We want to single out with special thanks, Nawoyuki Gregory Kubota and Wendy McKay for their special effort with the typesetting and the figures (including the cover illustration). We also thank the staff at Springer-Verlag, especially Achi Dosanjh, Laura Carlson, Ken Dreyhaupt and R¨diger Gebauer u for their skillful editorial work and production of the book. Jerry Marsden Pasadena, California Tudor Ratiu Santa Cruz, California Summer, 1998 ........................... 15 July 1998—18h02 ...........................

7. xii About the Authors Jerrold E. Marsden is Professor of Control and Dynamical Systems at Caltech. He got his B.Sc. at Toronto in 1965 and his Ph.D. from Princeton University in 1968, both in Applied Mathematics. He has done extensive research in mechanics, with applications to rigid body systems, fluid mechanics, elasticity theory, plasma physics as well as to general field theory. His primary current interests are in the area of dynamical systems and control theory, especially how it relates to mechanical systems with symmetry. He is one of the founders in the early 1970’s of reduction theory for mechanical systems with symmetry, which remains an active and much studied area of research today. He was the recipient of the prestigious Norbert Wiener prize of the American Mathematical Society and the Society for Industrial and Applied Mathematics in 1990, and was elected a fellow of the AAAS in 1997. He has been a Carnegie Fellow at Heriot–Watt Univer- sity (1977), a Killam Fellow at the University of Calgary (1979), recipient of the Jeffrey–Williams prize of the Canadian Mathematical Society in 1981, a Miller Professor at the University of California, Berkeley (1981–1982), a recipient of the Humboldt Prize in Germany (1991), and a Fairchild Fellow at Caltech (1992). He has served in several administrative capacities, such as director of the Research Group in Nonlinear Systems and Dynamics at Berkeley, 1984–86, the Advisory Panel for Mathematics at NSF, the Advisory committee of the Mathematical Sci- ences Institute at Cornell, and as Director of The Fields Institute, 1990–1994. He has served as an Editor for Springer-Verlag’s Applied Mathematical Sciences Se- ries since 1982 and serves on the editorial boards of several journals in mechanics, dynamics, and control. Tudor S. Ratiu is Professor of Mathematics at UC Santa Cruz and the Swiss Federal Institute of Technology in Lausanne. He got his B.Sc. in Mathematics and M.Sc. in Applied Mathematics, both at the University of Timi¸oara, Romania, s and his Ph.D. in Mathematics at Berkeley in 1980. He has previously taught at the University of Michigan, Ann Arbor, as a T. H. Hildebrandt Research Assis- tant Professor (1980–1983) and at the University of Arizona, Tucson (1983–1987). His research interests center on geometric mechanics, symplectic geometry, global analysis, and infinite dimensional Lie theory, together with their applications to integrable systems, nonlinear dynamics, continuum mechanics, plasma physics, and bifurcation theory. He has been a National Science Foundation Postdoctoral Fellow (1983–86), a Sloan Foundation Fellow (1984–87), a Miller Research Pro- fessor at Berkeley (1994), and a recipient of of the Humboldt Prize in Germany (1997). Since his arrival at UC Santa Cruz in 1987, he has been on the executive committee of the Nonlinear Sciences Organized Research Unit. He is currently managing editor of the AMS Surveys and Monographs series and on the editorial board of the Annals of Global Analysis and the Annals of the University of Timi¸oara. He was also a member of various research institutes such as MSRI in s Berkeley, the Center for Nonlinear Studies at Los Alamos, the Max Planck Insti- tute in Bonn, MSI at Cornell, IHES in Bures–sur–Yvette, The Fields Institute in Toronto (Waterloo), the Erwin Schro¨dinger Institute for Mathematical Physics o in Vienna, the Isaac Newton Institute in Cambridge, and RIMS in Kyoto. ........................... 15 July 1998—18h02 ...........................

8. This is page xiii Printer: Opaque this Contents Preface ix About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . xii I The Book xiv 1 Introduction and Overview 1 1.1 Lagrangian and Hamiltonian Formalisms . . . . . . . . . . . 1 1.2 The Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Lie–Poisson Brackets, Poisson Manifolds, Momentum Maps 9 1.4 The Heavy Top . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Incompressible Fluids . . . . . . . . . . . . . . . . . . . . . 18 1.6 The Maxwell–Vlasov System . . . . . . . . . . . . . . . . . 22 1.7 Nonlinear Stability . . . . . . . . . . . . . . . . . . . . . . . 29 1.8 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.9 The Poincar´–Melnikov Method . . . . . . . . . . . . e . . . . 46 1.10 Resonances, Geometric Phases, and Control . . . . . . . . . 49 2 Hamiltonian Systems on Linear Symplectic Spaces 61 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.2 Symplectic Forms on Vector Spaces . . . . . . . . . . . . . . 65 2.3 Canonical Transformations or Symplectic Maps . . . . . . . 69 2.4 The General Hamilton Equations . . . . . . . . . . . . . . . 73 2.5 When Are Equations Hamiltonian? . . . . . . . . . . . . . . 76

9. xiv Contents 2.6 Hamiltonian Flows . . . . . . . . . . . . . . . . . . . . . . . 80 2.7 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . 82 2.8 A Particle in a Rotating Hoop . . . . . . . . . . . . . . . . . 85 2.9 The Poincar´–Melnikov Method e and Chaos . . . . . . . . . 92 3 An Introduction to Infinite-Dimensional Systems 103 3.1 Lagrange’s and Hamilton’s Equations for Field Theory . . . 103 3.2 Examples: Hamilton’s Equations . . . . . . . . . . . . . . . 105 3.3 Examples: Poisson Brackets and Conserved Quantities . . . 113 4 Interlude: Manifolds, Vector Fields, and Differential Forms119 4.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.2 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . 126 4.3 The Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . 133 4.4 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . 137 5 Hamiltonian Systems on Symplectic Manifolds 143 5.1 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . 143 5.2 Symplectic Transformations . . . . . . . . . . . . . . . . . . 146 5.3 Complex Structures and K¨hler Manifolds . . . . a . . . . . . 148 5.4 Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . 153 5.5 Poisson Brackets on Symplectic Manifolds . . . . . . . . . . 156 6 Cotangent Bundles 161 6.1 The Linear Case . . . . . . . . . . . . . . . . . . . . . . . . 161 6.2 The Nonlinear Case . . . . . . . . . . . . . . . . . . . . . . 163 6.3 Cotangent Lifts . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.4 Lifts of Actions . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.5 Generating Functions . . . . . . . . . . . . . . . . . . . . . . 170 6.6 Fiber Translations and Magnetic Terms . . . . . . . . . . . 172 6.7 A Particle in a Magnetic Field . . . . . . . . . . . . . . . . 174 7 Lagrangian Mechanics 177 7.1 Hamilton’s Principle of Critical Action . . . . . . . . . . . . 177 7.2 The Legendre Transform . . . . . . . . . . . . . . . . . . . . 179 7.3 Euler–Lagrange Equations . . . . . . . . . . . . . . . . . . . 181 7.4 Hyperregular Lagrangians and Hamiltonians . . . . . . . . . 184 7.5 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.6 The Kaluza–Klein Approach to Charged Particles . . . . . . 196 7.7 Motion in a Potential Field . . . . . . . . . . . . . . . . . . 198 7.8 The Lagrange–d’Alembert Principle . . . . . . . . . . . . . 201 7.9 The Hamilton–Jacobi Equation . . . . . . . . . . . . . . . . 206 8 Variational Principles, Constraints, and Rotating Systems215 8.1 A Return to Variational Principles . . . . . . . . . . . . . . 215 ........................... 15 July 1998—18h02 ...........................

10. Contents xv 8.2 The Geometry of Variational Principles . . . . . . . . . . . 222 8.3 Constrained Systems . . . . . . . . . . . . . . . . . . . . . . 230 8.4 Constrained Motion in a Potential Field . . . . . . . . . . . 234 8.5 Dirac Constraints . . . . . . . . . . . . . . . . . . . . . . . . 238 8.6 Centrifugal and Coriolis Forces . . . . . . . . . . . . . . . . 244 8.7 The Geometric Phase for a Particle in a Hoop . . . . . . . . 249 8.8 Moving Systems . . . . . . . . . . . . . . . . . . . . . . . . 253 8.9 Routh Reduction . . . . . . . . . . . . . . . . . . . . . . . . 256 9 An Introduction to Lie Groups 261 9.1 Basic Definitions and Properties . . . . . . . . . . . . . . . 263 9.2 Some Classical Lie Groups . . . . . . . . . . . . . . . . . . . 279 9.3 Actions of Lie Groups . . . . . . . . . . . . . . . . . . . . . 308 10 Poisson Manifolds 329 10.1 The Definition of Poisson Manifolds . . . . . . . . . . . . . 329 10.2 Hamiltonian Vector Fields and Casimir Functions . . . . . . 335 10.3 Properties of Hamiltonian Flows . . . . . . . . . . . . . . . 340 10.4 The Poisson Tensor . . . . . . . . . . . . . . . . . . . . . . . 342 10.5 Quotients of Poisson Manifolds . . . . . . . . . . . . . . . . 355 10.6 The Schouten Bracket . . . . . . . . . . . . . . . . . . . . . 358 10.7 Generalities on Lie–Poisson Structures . . . . . . . . . . . . 365 11 Momentum Maps 371 11.1 Canonical Actions and Their Infinitesimal Generators . . . 371 11.2 Momentum Maps . . . . . . . . . . . . . . . . . . . . . . . . 373 11.3 An Algebraic Definition of the Momentum Map . . . . . . . 376 11.4 Conservation of Momentum Maps . . . . . . . . . . . . . . 378 11.5 Equivariance of Momentum Maps . . . . . . . . . . . . . . . 384 12 Computation and Properties of Momentum Maps 391 12.1 Momentum Maps on Cotangent Bundles . . . . . . . . . . . 391 12.2 Examples of Momentum Maps . . . . . . . . . . . . . . . . 396 12.3 Equivariance and Infinitesimal Equivariance . . . . . . . . . 404 12.4 Equivariant Momentum Maps Are Poisson . . . . . . . . . . 411 12.5 Poisson Automorphisms . . . . . . . . . . . . . . . . . . . . 420 12.6 Momentum Maps and Casimir Functions . . . . . . . . . . . 421 13 Lie–Poisson and Euler–Poincar´ Reduction e 425 13.1 The Lie–Poisson Reduction Theorem . . . . . . . . . . . . . 425 13.2 Proof of the Lie–Poisson Reduction Theorem for GL(n) . . 428 13.3 Proof of the Lie–Poisson Reduction Theorem for Diff vol (M ) 429 13.4 Lie–Poisson Reduction using Momentum Functions . . . . . 431 13.5 Reduction and Reconstruction of Dynamics . . . . . . . . . 433 13.6 The Linearized Lie–Poisson Equations . . . . . . . . . . . . 442 ........................... 15 July 1998—18h02 ...........................

11. xvi Contents 13.7 The Euler–Poincar´ Equations . . . . . . . . . . . . . . . . 445 e 13.8 The Lagrange–Poincar´ Equations . . . . . . . . . . . . . . 456 e 14 Coadjoint Orbits 459 14.1 Examples of Coadjoint Orbits . . . . . . . . . . . . . . . . . 460 14.2 Tangent Vectors to Coadjoint Orbits . . . . . . . . . . . . . 467 14.3 The Symplectic Structure on Coadjoint Orbits . . . . . . . 469 14.4 The Orbit Bracket via Restriction of the Lie–Poisson Bracket 475 14.5 The Special Linear Group on the Plane . . . . . . . . . . . 483 14.6 The Euclidean Group of the Plane . . . . . . . . . . . . . . 485 14.7 The Euclidean Group of Three-Space . . . . . . . . . . . . . 490 15 The Free Rigid Body 499 15.1 Material, Spatial, and Body Coordinates . . . . . . . . . . . 499 15.2 The Lagrangian of the Free Rigid Body . . . . . . . . . . . 501 15.3 The Lagrangian and Hamiltonian in Body Representation . 503 15.4 Kinematics on Lie Groups . . . . . . . . . . . . . . . . . . . 507 15.5 Poinsot’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 508 15.6 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . 511 15.7 The Hamiltonian of the Free Rigid Body in the Material Description via Euler Angles . . . . . . . . . . . . . . . . . 513 15.8 The Analytical Solution of the Free Rigid Body Problem . . 516 15.9 Rigid Body Stability . . . . . . . . . . . . . . . . . . . . . . 521 15.10Heavy Top Stability . . . . . . . . . . . . . . . . . . . . . . 525 15.11The Rigid Body and the Pendulum . . . . . . . . . . . . . . 529 ........................... 15 July 1998—18h02 ...........................

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15. This is page 1 Printer: Opaque this 1 Introduction and Overview 1.1 Lagrangian and Hamiltonian Formalisms Mechanics deals with the dynamics of particles, rigid bodies, continuous media (ﬂuid, plasma, and solid mechanics), and ﬁeld theories such as electromagnetism, gravity, etc. This theory plays a crucial role in quantum mechanics, control theory, and other areas of physics, engineering and even chemistry and biology. Clearly mechanics is a large subject that plays a fundamental role in science. Mechanics also played a key part in the development of mathematics. Starting with the creation of calculus stimulated by Newton’s mechanics, it continues today with exciting developments in group representations, geometry, and topology; these mathematical developments in turn are being applied to interesting problems in physics and engineering. Symmetry plays an important role in mechanics, from fundamental for- mulations of basic principles to concrete applications, such as stability criteria for rotating structures. The theme of this book is to emphasize the role of symmetry in various aspects of mechanics. This introduction treats a collection of topics fairly rapidly. The student should not expect to understand everything perfectly at this stage. We will return to many of the topics in subsequent chapters. Lagrangian and Hamiltonian Mechanics. Mechanics has two main points of view, Lagrangian mechanics and Hamiltonian mechanics. In one sense, Lagrangian mechanics is more fundamental since it is based on variational principles and it is what generalizes most directly to the

16. 2 1. Introduction and Overview general relativistic context. In another sense, Hamiltonian mechanics is more fundamental, since it is based directly on the energy concept and it is what is more closely tied to quantum mechanics. Fortunately, in many cases these branches are equivalent as we shall see in detail in Chapter 7. Needless to say, the merger of quantum mechanics and general relativity remains one of the main outstanding problems of mechanics. In fact, the methods of mechanics and symmetry are important ingredients in the developments of string theory that has attempted this merger. Lagrangian Mechanics. The Lagrangian formulation of mechanics is based on the observation that there are variational principles behind the fundamental laws of force balance as given by Newton’s law F = ma. One chooses a configuration space Q with coordinates q i , i = 1, . . . , n, that describe the configuration of the system under study. Then one introduces the Lagrangian L(q i , q i , t), which is shorthand notation for ˙ L(q 1 , . . . , q n , q 1 , . . . , q n , t). Usually, L is the kinetic minus the potential ˙ ˙ energy of the system and one takes q i = dq i /dt to be the system velocity. ˙ The variational principle of Hamilton states b δ L(q i , q i , t) dt = 0. ˙ (1.1.1) a In this principle, we choose curves q i (t) joining two fixed points in Q over a fixed time interval [a, b], and calculate the integral regarded as a function of this curve. Hamilton’s principle states that this function has a critical point at a solution within the space of curves. If we let δq i be a variation, that is, the derivative of a family of curves with respect to a parameter, then by the chain rule, (1.1.1) is equivalent to n b ∂L i ∂L i δq + i δ q dt = 0 ˙ (1.1.2) i=1 a ∂q i ∂q ˙ for all variations δq i . Using equality of mixed partials, one finds that d i δq . δqi = ˙ dt Using this, integrating the second term of (1.1.2) by parts, and employing the boundary conditions δq i = 0 at t = a and b, (1.1.2) becomes n b ∂L d ∂L − δq i dt = 0. (1.1.3) i=1 a ∂q i dt ∂ qi ˙ Since δq i is arbitrary (apart from being zero at the endpoints), (1.1.2) is equivalent to the Euler–Lagrange equations d ∂L ∂L − i = 0, i = 1, . . . , n. (1.1.4) dt ∂ q i ˙ ∂q ........................... 15 July 1998—18h02 ...........................

17. 1.1 Lagrangian and Hamiltonian Formalisms 3 As Hamilton himself realized around 1830, one can also gain valuable information by not imposing the fixed endpoint conditions. We will have a deeper look at such issues in Chapters 7 and 8. For a system of N particles moving in Euclidean 3-space, we choose the configuration space to be Q = R3N = R3 × · · · × R3 (N times) and L often has the form of kinetic minus potential energy: N 1 ˙ L(qi , qi , t) = ˙ mi qi 2 − V (qi ), (1.1.5) 2 i=1 where we write points in Q as q1 , . . . , qN , where qi ∈ R3 . In this case the Euler–Lagrange equations (1.1.4) reduce to Newton’s second law d ∂V (mi qi ) = − ˙ ; i = 1, . . . , N (1.1.6) dt ∂qi that is, F = ma for the motion of particles in the potential field V . As we shall see later, in many examples more general Lagrangians are needed. Generally, in Lagrangian mechanics, one identifies a configuration space Q (with coordinates q 1 , . . . , q n )) and then forms the velocity phase space T Q also called the tangent bundle of Q. Coordinates on T Q are denoted (q 1 , . . . , q n , q 1 , . . . , q n ), ˙ ˙ and the Lagrangian is regarded as a function L : T Q → R. Already at this stage, interesting links with geometry are possible. If gij (q) is a given metric tensor or mass matrix (for now, just think of this as a q-dependent positive-definite symmetric n×n matrix) and we consider the kinetic energy Lagrangian n 1 L(q i , q i ) = ˙ gij (q)q i q j , ˙ ˙ (1.1.7) 2 i,j=1 then the Euler–Lagrange equations are equivalent to the equations of geodesic motion, as can be directly verified (see §7.5 for details). Conservation laws that are a result of symmetry in a mechanical context can then be applied to yield interesting geometric facts. For instance, theorems about geodesics on surfaces of revolution can be readily proved this way. The Lagrangian formalism can be extended to the infinite dimensional case. One view (but not the only one) is to replace the q i by fields ϕ1 , . . . , ϕm which are, for example, functions of spatial points xi and time. Then L is a function of ϕ1 , . . . , ϕm , ϕ1 , . . . , ϕm and the spatial derivatives of the ˙ ˙ fields. We shall deal with various examples of this later, but we emphasize that properly interpreted, the variational principle and the Euler–Lagrange equations remain intact. One replaces the partial derivatives in the Euler– Lagrange equations by functional derivatives defined below. ........................... 15 July 1998—18h02 ...........................

18. 4 1. Introduction and Overview Hamiltonian Mechanics. To pass to the Hamiltonian formalism, introduce the conjugate momenta ∂L pi = , i = 1, . . . , n, (1.1.8) ∂ qi ˙ make the change of variables (q i , q i ) → (q i , pi ), and introduce the Hamil- ˙ tonian n H(q i , pi , t) = pj q j − L(q i , q i , t). ˙ ˙ (1.1.9) j=1 Remembering the change of variables, we make the following computations using the chain rule: n ∂H ∂ qj ˙ ∂L ∂ q j ˙ = qi + ˙ pj − j = qi ˙ (1.1.10) ∂pi j=1 ∂pi ∂ q ∂pi ˙ and n n ∂H ∂ qj ˙ ∂L ∂L ∂ q j ˙ ∂L = pj − i− = − i, (1.1.11) ∂q i j=1 ∂q i ∂q j=1 ∂ q j ∂q i ˙ ∂q where (1.1.8) has been used twice. Using (1.1.4) and (1.1.8), we see that (1.1.11) is equivalent to ∂H d = − pi . (1.1.12) ∂q i dt Thus, the Euler–Lagrange equations are equivalent to Hamilton’s equations dq i ∂H = , dt ∂pi (1.1.13) dpi ∂H =− i, dt ∂q where i = 1, . . . , n. The analogous Hamiltonian partial diﬀerential equations for time dependent ﬁelds ϕ1 , . . . , ϕm and their conjugate momenta π1 , ..., πm , are ∂ϕa δH = ∂t δπa (1.1.14) ∂πa δH = − a, ∂t δϕ ........................... 15 July 1998—18h02 ...........................

19. 1.1 Lagrangian and Hamiltonian Formalisms 5 where a = 1, . . . , m, and H is a functional of the fields ϕa and πa , and the variational or functional derivatives are defined by the equation δH 1 n 1 δϕ d x = lim [H(ϕ1 +εδϕ1 , ϕ2 , . . . , ϕm , π1 , . . . , πm ) Rn δϕ1 ε→0 ε − H(ϕ1 , ϕ2 , . . . , ϕm , π1 , . . . , πm )], (1.1.15) and similarly for δH/δϕ2 , . . . , δH/δπm . Equations (1.1.13) and (1.1.14) can be recast in Poisson bracket form ˙ F = {F, H}, (1.1.16) where the brackets in the respective cases are given by n ∂F ∂G ∂F ∂G {F, G} = − (1.1.17) i=1 ∂q i ∂pi ∂pi ∂q i and m δF δG δF δG {F, G} = − dn x. (1.1.18) a=1 Rn δϕa δπa δπa δϕa Associated to any configuration space Q (coordinatized by (q 1 , . . . , q n )) is a phase space T ∗ Q called the cotangent bundle of Q, which has coordinates (q 1 , . . . , q n , p1 , . . . , pn ). On this space, the canonical bracket (1.1.17) is intrinsically defined in the sense that the value of {F, G} is independent of the choice of coordinates. Because the Poisson bracket satisfies {F, G} = −{G, F } and in particular {H, H} = 0 , we see from (1.1.16) ˙ that H = 0; that is, energy is conserved . This is the most elementary of many deep and beautiful conservation properties of mechanical systems. There is also a variational principle on the Hamiltonian side. For the Euler–Lagrange equations, we deal with curves in q-space (configuration space), whereas for Hamilton’s equations we deal with curves in (q, p)-space (momentum phase space). The principle is b n δ [pi q i − H(q j , pj )] dt = 0 ˙ (1.1.19) a i=1 as is readily verified; one requires pi δq i = 0 at the endpoints. This formalism is the basis for the analysis of many important systems in particle dynamics and field theory, as described in standard texts such as Whittaker [1927], Goldstein [1980], Arnold [1989], Thirring [1978], and Abraham and Marsden [1978]. The underlying geometric structures that are important for this formalism are those of symplectic and Poisson geometry. ........................... 15 July 1998—18h02 ...........................

20. 6 1. Introduction and Overview How these structures are related to the Euler–Lagrange equations and variational principles via the Legendre transformation is an essential ingredient of the story. Furthermore, in the infinite-dimensional case it is fairly well understood how to deal rigorously with many of the functional analytic difficulties that arise; see, for example, Chernoff and Marsden [1974] and Marsden and Hughes [1983]. Exercises 1.1-1. Show by direct calculation that the classical Poisson bracket satisfies the Jacobi identity . That is, if F and K are both functions of the 2n variables (q 1 , q 2 , . . . , q n , p1 , p2 , ..., pn ) and we define n ∂F ∂K ∂K ∂F {F, K} = − i , i=1 ∂q i ∂pi ∂q ∂pi then the identity {L, {F, K}} + {K, {L, F }} + {F, {K, L}} = 0 holds. 1.2 The Rigid Body It was already clear in the last century that certain mechanical systems resist the canonical formalism outlined in §1.1. For example, to obtain a Hamiltonian description for fluids, Clebsch [1857, 1859] found it necessary to introduce certain nonphysical potentials1 . We will discuss fluids in §1.4 below. Euler’s Rigid Body Equations. In the absence of external forces, the Euler equations for the rotational dynamics of a rigid body about its center of mass are usually written as follows, as we shall derive in detail in Chapter 15: ˙ I1 Ω1 = (I2 − I3 )Ω2 Ω3 , ˙ I2 Ω2 = (I3 − I1 )Ω3 Ω1 , (1.2.1) ˙ I3 Ω3 = (I1 − I2 )Ω1 Ω2 , where Ω = (Ω1 , Ω2 , Ω3 ) is the body angular velocity vector (the angular velocity of the rigid body as seen from a frame fixed in the body) and I1 , I2 , I3 are constants depending on the shape and mass distribution of the body—the principal moments of inertia of the rigid body. 1 For a geometric account of Clebsch potentials and further references, see Marsden and Weinstein [1983], Marsden, Ratiu, and Weinstein [1984a,b], Cendra and Marsden [1987], and Cendra, Ibort, and Marsden [1987]. ........................... 15 July 1998—18h02 ...........................

21. 1.2 The Rigid Body 7 Are equations (1.2.1) Lagrangian or Hamiltonian in any sense? Since there are an odd number of equations, they obviously cannot be put in canonical Hamiltonian form in the sense of equations (1.1.13). A classical way to see the Lagrangian (or Hamiltonian) structure of the rigid body equations is to use a description of the orientation of the body ˙ ˙ ˙ in terms of three Euler angles denoted θ, ϕ, ψ and their velocities θ, ϕ, ψ (or conjugate momenta pθ , pϕ , pψ ), relative to which the equations are in Euler–Lagrange (or canonical Hamiltonian) form. However, this procedure requires using six equations while many questions are easier to study using the three equations (1.2.1). Lagrangian Form. To see the sense in which (1.2.1) are Lagrangian, introduce the Lagrangian 1 L(Ω) = (I1 Ω2 + I2 Ω2 + I3 Ω2 ) 1 2 3 (1.2.2) 2 which, as we will see in detail in Chapter 15, is the (rotational) kinetic energy of the rigid body. Regarding IΩ = (I1 Ω1 , I2 Ω2 , I3 Ω3 ) as a vector, write (1.2.1) as d ∂L ∂L = × Ω. (1.2.3) dt ∂Ω ∂Ω These equations appear explicitly in Lagrange [1788] (Volume 2, p.212) and were generalized to arbitrary Lie algebras by Poincar´ [1901b]. We will e discuss these general Euler-Poincar´ equations in Chapter 13. We can e also write a variational principle for (1.2.3) that is analogous to that for the Euler–Lagrange equations, but is written directly in terms of Ω. Namely, (1.2.3) is equivalent to b δ L dt = 0, (1.2.4) a where variations of Ω are restricted to be of the form ˙ δΩ = Σ + Ω × Σ, (1.2.5) where Σ is a curve in R3 that vanishes at the endpoints. This may be proved in the same way as we proved that the variational principle (1.1.1) is equivalent to the Euler–Lagrange equations (1.1.4); see Exercise 1.2-2. In fact, later on in Chapter 13, we shall see how to derive this variational principle from the more “primitive” one (1.1.1). Hamiltonian Form. If, instead of variational principles, we concentrate on Poisson brackets and drop the requirement that they be in the canonical form (1.1.17), then there is also a simple and beautiful Hamiltonian ........................... 15 July 1998—18h02 ...........................

22. 8 1. Introduction and Overview structure for the rigid body equations. To state it, introduce the angular momenta ∂L Πi = Ii Ωi = , i = 1, 2, 3, (1.2.6) ∂Ωi so that the Euler equations become ˙ I2 − I3 Π1 = Π2 Π3 , I2 I3 ˙ I3 − I1 Π2 = Π3 Π1 , (1.2.7) I3 I1 ˙ I1 − I2 Π3 = Π1 Π2 , I1 I2 that is, ˙ Π = Π × Ω. (1.2.8) Introduce the following rigid body Poisson bracket on functions of the Π’s: {F, G}(Π) = −Π · ( F × G) (1.2.9) and the Hamiltonian 1 Π2 1 Π2 Π2 H= + 2+ 3 . (1.2.10) 2 I1 I2 I3 One checks (Exercise 1.2-3) that Euler’s equations (1.2.7) are equivalent to2 ˙ F = {F, H}. (1.2.11) For any equation of the form (1.2.11), conservation of total angular momentum holds regardless of the Hamiltonian; indeed, with 1 2 C(Π) = (Π + Π2 + Π2 ), 2 1 2 3 we have C(Π) = Π, and so d 1 2 (Π + Π2 + Π2 ) = {C, H}(Π) dt 2 1 2 3 = −Π · ( C × H) = −Π · (Π × H) = 0. 2 This simple result is implicit in many works, such as Arnold [1966, 1969], and is given explicitly in this form for the rigid body in Sudarshan and Mukunda [1974]. (Some preliminary versions were given by Pauli [1953], Martin [1959], and Nambu [1973].) On the other hand, the variational form (1.2.4) appears to be due to Poincar´ [1901b] and e Hamel [1904], at least implicitly. It is given explicitly for ﬂuids in Newcomb [1962] and Bretherton [1970] and in the general case in Marsden and Scheurle [1993a,b]. ........................... 15 July 1998—18h02 ...........................

23. 1.2 The Rigid Body 9 The same calculation shows that {C, F } = 0 for any F . Functions such as these that Poisson commute with every function are called Casimir functions; they play an important role in the study of stability, as we shall see later3 . Exercises 1.2-1. Show by direct calculation that the rigid body Poisson bracket satisfies the Jacobi identity. That is, if F and K are both functions of (Π1 , Π2 , Π3 ) and we define {F, K}(Π) = −Π · ( F × K), then the identity {L, {F, K}} + {K, {L, F }} + {F, {K, L}} = 0 holds. 1.2-2. Verify directly that the Euler equations for a rigid body are equivalent to δ L dt = 0 ˙ for variations of the form δΩ = Σ + Ω × Σ, where Σ vanishes at the endpoints. 1.2-3. Verify directly that the Euler equations for a rigid body are equivalent to the equations d F = {F, H}, dt where { , } is the rigid body Poisson bracket and H is the rigid body Hamil- tonian. 1.2-4. (a) Show that the rotation group SO(3) can be identified with the Poin- car´ sphere: that is, the unit circle bundle of the two sphere S 2 , e defined to be the set of unit tangent vectors to the two-sphere in R3 . (b) Using the known fact from basic topolgy that any (continuous) vector field on S 2 must vanish somewhere, show that SO(3) cannot be written as S 2 × S 1 . 3 H. B. G. Casimir was a student of P. Ehrenfest and wrote a brilliant thesis on the quantum mechanics of the rigid body, a problem that has not been adequately addressed in the detail that would be desirable, even today. Ehrenfest in turn wrote his thesis under Boltzmann around 1900 on variational principles in fluid dynamics and was one of the first to study fluids from this point of view in material, rather than Clebsch representation. Curiously, Ehrenfest used the Gauss–Hertz principle of least curvature rather than the more elementary Hamilton prinicple. This is a seed for many important ideas in this book. ........................... 15 July 1998—18h02 ...........................

24. 10 1. Introduction and Overview 1.3 Lie–Poisson Brackets, Poisson Manifolds, Momentum Maps The rigid body variational principle and the rigid body Poisson bracket are special cases of general constructions associated to any Lie algebra g, that is, a vector space together with a bilinear, antisymmetric bracket [ξ, η] satisfying Jacobi’s identity : [[ξ, η], ζ] + [[ζ, ξ], η] + [[η, ζ], ξ] = 0 (1.3.1) for all ξ, η, ζ ∈ g. For example, the Lie algebra associated to the rotation group is g = R3 with bracket [ξ, η] = ξ × η, the ordinary vector cross product. The Euler-Poincar´ Equations. e The construction of a variational principle on g, replaces ˙ δΩ = Σ + Ω × Σ by δξ = η + [η, ξ]. ˙ The resulting general equations on g, which we will study in detail in Chap- ter 13, are called the Euler-Poincar´ equations. These equations are e valid for either finite or infinite dimensional Lie algebras. To state them in the finite dimensional case, we use the following notation. Choosing a basis d e1 , . . . , er of g (so dim g = r), the structure constants Cab are defined by the equation r d [ea , eb ] = Cab ed , (1.3.2) d=1 where a, b run from 1 to r. If ξ is an element of the Lie algebra, its components relative to this basis are denoted ξ a . If e1 , . . . , er is the corresponding dual basis, then the components of the differential of the Lagrangian L are the partial derivatives ∂L/∂ξ a . Then the Euler-Poincar´ equations are e r d ∂L b ∂L a = Cad ξ . (1.3.3) dt ∂ξ d ∂ξ b a,b=1 The coordinate-free version reads d ∂L ∂L = ad∗ ξ , dt ∂ξ ∂ξ where adξ : g → g is the linear map η → [ξ, η] and ad∗ : g∗ → g∗ is its ξ dual. For example, for L : R3 → R, the Euler-Poincar´ equations become e d ∂L ∂L = × Ω, dt ∂Ω ∂Ω ........................... 15 July 1998—18h02 ...........................

25. 1.3 Lie–Poisson Brackets, Poisson Manifolds, Momentum Maps 11 which generalize the Euler equations for rigid body motion. As we mentioned earlier, these equations were written down for a fairly general class of L by Lagrange [1788, Volume 2, Equation A on p. 212], while it was Poincar´ [1901b] who generalized them to any Lie algebra. e The generalization of the rigid body variational principle states that the Euler–Poincar´ equations are equivalent to e δ L dt = 0 (1.3.4) for all variations of the form δξ = η + [ξ, η] for some curve η in g that ˙ vanishes at the end points. The Lie–Poisson Equations. We can also generalize the rigid body Poisson bracket as follows: Let F, G be defined on the dual space g∗ . De- noting elements of g∗ by µ, let the functional derivative of F at µ be the unique element δF/δµ of g defined by 1 δF lim [F (µ + εδµ) − F (µ)] = δµ, , (1.3.5) ε→0ε δµ for all δµ ∈ g∗ , where , denotes the pairing between g∗ and g. This definition (1.3.5) is consistent with the definition of δF/δϕ given in (1.1.15) when g and g∗ are chosen to be appropriate spaces of fields. Define the (±) Lie–Poisson brackets by δF δG {F, G}± (µ) = ± µ, , . (1.3.6) δµ δµ Using the coordinate notation introduced above, the (±) Lie–Poisson brackets become r ∂F ∂G {F, G}± (µ) = ± d Cab µd , (1.3.7) ∂µa ∂µb a,b,d=1 where µ = µa ea . Poisson Manifolds. The Lie–Poisson bracket and the canonical brackets from the last section have four simple but crucial properties: PB1 {F, G} is real bilinear in F and G. PB2 {F, G} = −{G, F }, antisymmetry. PB3 {F, G}, H} + {{H, F }, G} + {{G, H}, F } = 0, Jacobi identity. PB4 {F G, H} = F {G, H} + {F, H}G, Leibniz identity. A manifold (that is, an n-dimensional “smooth surface”) P together with a bracket operation on F(P ), the space of smooth functions on P , ........................... 15 July 1998—18h02 ...........................

26. 12 1. Introduction and Overview and satisfying properties PB1–PB4, is called a Poisson manifold . In particular, g∗ is a Poisson manifold . In Chapter 10 we will study the general concept of a Poisson manifold. For example, if we choose g = R3 with the bracket taken to be the cross product [x, y] = x × y, and identify g∗ with g using the dot product on R3 (so Π, x = Π · x is the usual dot product), then the (−) Lie–Poisson bracket becomes the rigid body bracket. Hamiltonian Vector Fields. On a Poisson manifold (P, {· , ·}), associated to any function H there is a vector field, denoted by XH , which has the property that for any smooth function F : P → R we have the identity dF, XH = dF · XH = {F, H}. where dF is the differential fo F . We say that the vector field XH is generated by the function H or that XH is the Hamiltonian vector field associated with H. We also define the associated dynamical system whose points z in phase space evolve in time by the differential equation z = XH (z). ˙ (1.3.8) This definition is consistent with the equations in Poisson bracket form (1.1.16). The function H may have the interpretation of the energy of the system, but of course the definition (1.3.8) makes sense for any function. For canonical systems with the Poisson bracket given by (1.1.17), XH is given by the formula ∂H ∂H XH (q i , pi ) = ,− i , (1.3.9) ∂pi ∂q whereas for the rigid body bracket given on R3 by (1.2.9), XH (Π) = Π × H(Π). (1.3.10) ˙ The general Lie–Poisson equations, determined by F = {F, H} read r d ∂H µa = ˙ µd Cab , ∂µb b,c=1 or intrinsically, µ= ˙ ad∗ δH/δµ µ. (1.3.11) Reduction. There is an important feature of the rigid body bracket that also carries over to more general Lie algebras, namely, Lie–Poisson brackets arise from canonical brackets on the cotangent bundle (phase space) T ∗ G associated with a Lie group G which has g as its associated Lie algebra. ........................... 15 July 1998—18h02 ...........................

27. 1.3 Lie–Poisson Brackets, Poisson Manifolds, Momentum Maps 13 (The general theory of Lie groups is presented in Chapter 9.) Specifically, there is a general construction underlying the association (θ, ϕ, ψ, pθ , pϕ , pψ ) → (Π1 , Π2 , Π3 ) (1.3.12) defined by: 1 Π1 = [(pϕ − pψ cos θ) sin ψ + pθ sin θ cos ψ], sin θ 1 (1.3.13) Π2 = [(pϕ − pψ cos θ) cos ψ − pθ sin θ sin ψ], sin θ Π 3 = pψ . This rigid body map takes the canonical bracket in the variables (θ, ϕ, ψ) and their conjugate momenta (pθ , pϕ , pψ ) to the (−) Lie–Poisson bracket in the following sense. If F and K are functions of Π1 , Π2 , Π3 , they determine functions of (θ, ϕ, ψ, pθ , pϕ , pψ ) by substituting (1.3.13). Then a (tedious but straightforward) exercise using the chain rule shows that {F, K}(−){Lie-Poisson} = {F, K}canonical . (1.3.14) We say that the map defined by (1.3.13) is a canonical map or a Poisson map and that the (−) Lie–Poisson bracket has been obtained from the canonical bracket by reduction. For a rigid body free to rotate about is center of mass, G is the (proper) rotation group SO(3) and the Euler angles and their conjugate momenta are coordinates for T ∗ G. The choice of T ∗ G as the primitive phase space is made according to the classical procedures of mechanics: the configuration space SO(3) is chosen since each element A ∈ SO(3) describes the orientation of the rigid body relative to a reference configuration, that is, the rotation A maps the reference configuration to the current configuration. For the description using Lagrangian mechanics, one forms the velocity- ˙ ˙ ˙ phase space T SO(3) with coordinates (θ, ϕ, ψ, θ, ϕ, ψ). The Hamiltonian description is obtained as in §1.1 by using the Legendre transform which maps T G to T ∗ G. The passage from T ∗ G to the space of Π’s (body angular momentum space) given by (1.3.13) turns out to be determined by left translation on the group. This mapping is an example of a momentum map; that is, a mapping whose components are the “Noether quantities” associated with a symmetry group. The map (1.3.13) being a Poisson (canonical) map (see equation (1.3.14)) is a general fact about momentum maps proved in §12.6. To get to space coordinates one would use right translations and the (+) bracket. This is what is done to get the standard description of fluid dynamics. Momentum Maps and Coadjoint Orbits. From the general rigid ˙ body equations, Π = Π × H, we see that d 2 Π = 0. dt ........................... 15 July 1998—18h02 ...........................

28. 14 1. Introduction and Overview In other words, Lie–Poisson systems on R3 conserve the total angular momenta; that is, leave the spheres in Π-space invariant. The generalization of these objects associated to arbitrary Lie algebras are called coadjoint orbits. Coadjoint orbits are submanifolds of g∗ , with the property that any Lie– ˙ Poisson system F = {F, H} leaves them invariant. We shall also see how these spaces are Poisson manifolds in their own right and are related to the right (+) or left (−) invariance of the system regarded on T ∗ G, and the corresponding conserved Noether quantities. On a general Poisson manifold (P, {· , ·}), the definition of a momentum map is as follows. We assume that a Lie group G with Lie algebra g acts on P by canonical transformations. As we shall review later (see Chapter 9), the infinitesimal way of specifying the action is to associate to each Lie algebra element ξ ∈ g a vector field ξP on P . A momentum map is a map J : P → g∗ with the property that for every ξ ∈ g, the function J, ξ (the pairing of the g∗ valued function J with the vector ξ) generates the vector field ξP ; that is, X J,ξ = ξp . As we shall see later, this definition generalizes the usual notions of linear and angular momentum. The rigid body shows that the notion has much wider interest. A fundamental fact about momentum maps is that if the Hamiltonian H is invariant under the action of the group G, then the vector valued function J is a constant of the motion for the dynamics of the Hamiltonian vector field XH associated to H. One of the important notions related to momentum maps is that of infinitesimal equivariance or the classical commutation relations, which state that { J, ξ , J, η } = J, [ξ, η] (1.3.15) for all Lie algebra elements ξ and η. Relations like this are well known for the angular momentum, and can be directly checked using the Lie algebra of the rotation group. Later, in Chapter 12 we shall see that the relations (1.3.15) hold for a large important class of momentum maps that are given by computable formulas. Remarkably, it is the condition (1.3.15) that is exactly what is needed to prove that J is, in fact, a Poisson map. It is via this route that one gets an intellectually satisfying generalization of the fact that the map defined by equations (1.3.13) is a Poisson map, that is, equation (1.3.14) holds. Some History. The Lie–Poisson bracket was discovered by Sophus Lie (Lie [1890], Vol. II, p. 237). However, Lie’s bracket and his related work was not given much attention until the work of Kirillov, Kostant, and Souriau (and others) revived it in the mid-1960s. Meanwhile, it was noticed by Pauli and Martin around 1950 that the rigid body equations are in Hamiltonian ........................... 15 July 1998—18h02 ...........................

29. 1.3 Lie–Poisson Brackets, Poisson Manifolds, Momentum Maps 15 form using the rigid body bracket, but they were apparently unaware of the underlying Lie theory. Meanwhile, the generalization of the Euler equations to any Lie algebra g by Poincar´ [1901b] (and picked up by Hamel [1904]) e proceeded as well, but without much contact with Lie’s work until recently. The symplectic structure on coadjoint orbits also has a complicated history and itself goes back to Lie (Lie [1890], Ch. 20). The general notion of a Poisson manifold also goes back to Lie, However, the four defining properties of the Poisson bracket have been isolated by many authors such as Dirac [1964], p. 10. The term “Poisson manifold” was coined by Lichnerowicz [1977]. We shall give more historical information on Poisson manifolds in §10.3. The notion of the momentum map (the English translation of the French words “application moment”) also has roots going back to the work of Lie.4 Momentum maps have found an astounding array of applications beyond those already mentioned. For instance, they are used in the study of the space of all solutions of a relativistic field theory (see Arms, Marsden and Moncrief [1982]) and in the study of singularities in algebraic geometry (see Atiyah [1983] and Kirwan [1984a]). They also enter into convex analysis in many interesting ways, such as the Schur-Horn theorem (Schur [1923], Horn [1954]) and its generalizations (Kostant [1973]) and in the theory of integrable systems (Bloch, Brockett, and Ratiu [1990, 1992] and Bloch, Flaschka, and Ratiu [1990, 1993]). It turns out that the image of the momentum map has remarkable convexity properties: see Atiyah [1982], Guillemin and Sternberg [1982, 1984], Kirwan [1984b], Delzant [1988], Lu and Ratiu [1991], Sjamaar [1996], and Flaschka and Ratiu [1997]. Exercises 1.3-1. A linear operator D on the space of smooth functions on Rn is called a derivation if it satisfies the Leibniz identity: D(F G) = (DF )G + F (DG). Accept the fact from the theory of manifolds (see Chapter 4) that in local coordinates the expression of DF takes the form n ∂F (DF )(x) = ai (x) (x) i=1 ∂xi for some smooth functions a1 , . . . , an . 4 Many authors use the words “moment map” for what we call the “momentum map.” In English, unlike French, one does not use the phrases “linear moment” or “angular moment of a particle”, and correspondingly we prefer to use “momentum map.” We shall give some comments on the history of momentum maps in §11.2. ........................... 15 July 1998—18h02 ...........................

30. 16 1. Introduction and Overview (a) Use the fact just stated to prove that for any Poisson bracket { , } on Rn , we have n ∂F ∂G {F, G} = {xi , xj } . i,j=1 ∂xi ∂xj (b) Show that the Jacobi identity holds for a Poisson bracket { , } on Rn if and only if it holds for the coordinate functions. 1.3-2. (a) Define, for a fixed function f : R3 → R {F, K}f = f ·( F × K). Show that this is a Poisson bracket. (b) Locate the bracket in part (a) in Nambu [1973]. 1.3-3. Verify directly that (1.3.13) defines a Poisson map. 1.3-4. Show that a bracket satisfying the Leibniz identity also satisfies F {K, L} − {F K, L} = {F, K}L − {F, KL}. 1.4 The Heavy Top The equations of motion for a rigid body with a fixed point in a gravitational field provide another interesting example of a system which is Hamiltonian relative to a Lie–Poisson bracket. See Figure 1.4.1. The underlying Lie algebra consists of the algebra of infinitesimal Eu- clidean motions in R3 . (These do not arise as Euclidean motions of the body since the body has a fixed point). As we shall see, there is a close parallel with the Poisson structure for compressible fluids. The basic phase space we start with is again T ∗ SO(3), coordinatized by Euler angles and their conjugate momenta. In these variables, the equations are in canonical Hamiltonian form; however, the presence of gravity breaks the symmetry and the system is no longer SO(3) invariant, so it cannot be written entirely in terms of the body angular momentum Π. One also needs to keep track of Γ, the “direction of gravity” as seen from the body. This is defibed by Γ = A−1 k, where k points upward and A is the element of SO(3) describing the current configuration of the body. The equations of motion are ˙ I2 − I3 Π1 = Π2 Π3 + M gl(Γ2 χ3 − Γ3 χ2 ), I2 I3 ˙ I3 − I1 Π2 = Π3 Π1 + M gl(Γ3 χ1 − Γ1 χ3 ), (1.4.1) I3 I1 ˙ I1 − I2 Π3 = Π1 Π2 + M gl(Γ1 χ2 − Γ2 χ1 ) I1 I2 ........................... 15 July 1998—18h02 ...........................

31. 1.4 The Heavy Top 17 Ω M = total mass g = gravitational acceleration center of mass Ω = body angular velocity of top l = distance from fixed point to center of mass lAχ g fixed point k Γ Figure 1.4.1. Heavy top and ˙ Γ=Γ×Ω (1.4.2) where M is the body’s mass, g is the acceleration of gravity, χ is the body ﬁxed unit vector on the line segment connecting the ﬁxed point with the body’s center of mass, and l is the length of this segment. See Figure 1.4.1. The Lie algebra of the Euclidean group is se(3) = R3 × R3 with the Lie bracket [(ξ, u), (η, v)] = (ξ × η, ξ × v − η × u). (1.4.3) We identify the dual space with pairs (Π, Γ); the corresponding (−) Lie– Poisson bracket, called the heavy top bracket, is {F, G}(Π, Γ) = −Π · ( ΠF × Π G) −Γ·( ΠF × ΓG − ΠG × Γ F ). (1.4.4) The above equations for Π, Γ can be checked to be equivalent to ˙ F = {F, H}, (1.4.5) where the heavy top Hamiltonian 1 Π2 Π2 Π2 H(Π, Γ) = 1 + 2+ 3 + M glΓ · χ (1.4.6) 2 I1 I2 I3 ........................... 15 July 1998—18h02 ...........................

32. 18 1. Introduction and Overview is the total energy of the body (Sudarshan and Mukunda [1974]). The Lie algebra of the Euclidean group has a structure which is a special case of what is called a semidirect product. Here it is the product of the group of rotations with the translation group. It turns out that semidirect products occur under rather general circumstances when the symmetry in T ∗ G is broken. In particular, notice the similarities in structure between the Poisson bracket (1.6.16) for compressible flow and (1.4.4). For compressible flow it is the density which prevents a full Diff(Ω) invariance; the Hamiltonian is only invariant under those diffeomorphisms that preserve the density. The general theory for semidirect products was developed by Sudarshan and Mukunda [1974], Ratiu [1980, 1981, 1982], Guillemin and Sternberg [1982], Marsden, Weinstein, Ratiu, Schmid, and Spencer [1983], Marsden, Ratiu, and Weinstein [1984a,b], and Holm and Kupersh- midt [1983]. The Lagrangian approach to this and related problems is given in Holm, Marsden, and Ratiu [1998]. Exercises ˙ 1.4-1. Verify that F = {F, H} are equivalent to the heavy top equations using the heavy top Hamiltonian and bracket. 1.4-2. Work out the Euler–Poincar´ equations on se(3). Show that with e L(Ω, Γ) = 1 (I1 Ω2 + I2 Ω2 + I3 Ω2 ) − M glΓ · χ, the Euler–Poincar´ equations 2 1 2 3 e are not the heavy top equations. 1.5 Incompressible Fluids Arnold [1966a, 1969] showed that the Euler equations for an incompressible fluid could be given a Lagrangian and Hamiltonian description similar to that for the rigid body. His approach5 has the appealing feature that one sets things up just the way Lagrange and Hamilton would have done: one begins with a configuration space Q, forms a Lagrangian L on the velocity phase space T Q and then H on the momentum phase space T ∗ Q, just as was outlined in §1.1. Thus, one automatically has variational principles, etc. For ideal fluids, Q = G is the group Diff vol (Ω) of volume preserving transformations of the fluid container (a region Ω in R2 or R3 , or a Rieman- nian manifold in general, possibly with boundary). Group multiplication in G is composition. Kinematics of a Fluid. The reason we select G = Diff vol (Ω) as the configuration space is similar to that for the rigid body; namely, each ϕ 5 Arnold’s approach is consistent with what appears in the thesis of Ehrenfest from around 1904; see Klein [1970]. However, Ehrenfest bases his principles on the more sophisticated curvature principles of Gauss and Hertz. ........................... 15 July 1998—18h02 ...........................

33. 1.5 Incompressible Fluids 19 in G is a mapping of Ω to Ω which takes a reference point X ∈ Ω to a current point x = ϕ(X) ∈ Ω; thus, knowing ϕ tells us where each particle of fluid goes and hence gives us the fluid configuration. We ask that ϕ be a diffeomorphism to exclude discontinuities, cavitation, and fluid inter- penetration, and we ask that ϕ be volume preserving to correspond to the assumption of incompressibility. A motion of a fluid is a family of time-dependent elements of G, which we write as x = ϕ(X, t). The material velocity field is defined by ∂ϕ(X, t) V(X, t) = , ∂t and the spatial velocity field is defined by v(x, t) = V(X, t), where x and X are related by x = ϕ(X, t). If we suppress “t” and write ϕ for V, note ˙ that v = ϕ ◦ ϕ−1 ˙ i.e., vt = Vt ◦ ϕ−1 , t (1.5.1) where ϕt (x) = ϕ(X, t). See Figure 1.5.1. trajectory of fluid particle D u(x,t) Figure 1.5.1. We can regard (1.5.1) as a map from the space of (ϕ, ϕ) (material or La- ˙ grangian description) to the space of v’s (spatial or Eulerian description). Like the rigid body, the material to spatial map (1.5.1) takes the canonical bracket to a Lie–Poisson bracket; one of our goals is to understand this reduction. Notice that if we replace ϕ by ϕ ◦ η for a fixed (time-independent) η ∈ Diff vol (Ω), then ϕ ◦ ϕ−1 is independent of η; this reflects the right ˙ invariance of the Eulerian description (v is invariant under composition of ϕ by η on the right). This is also called the particle relabeling symmetry of fluid dynamics. The spaces T G and T ∗ G represent the Lagrangian (material) description and we pass to the Eulerian (spatial) description by right translations and use the (+) Lie–Poisson bracket. One of the things we want to do later is to better understand the reason for the switch between right and left in going from the rigid body to fluids. ........................... 15 July 1998—18h02 ...........................

34. 20 1. Introduction and Overview Dynamics of a Fluid. The Euler equations for an ideal, incompressible, homogeneous fluid moving in the region Ω are ∂v + (v · )v = − p (1.5.2) ∂t with the constraint div v = 0 and the boundary conditions: v is tangent to the boundary, ∂Ω. The pressure p is determined implicitly by the divergence-free (volume preserving) constraint div v = 0. (See Chorin and Marsden [1993] for basic information on the derivation of Euler’s equations.) The associated Lie algebra g is the space of all divergence-free vector fields tangent to the boundary. This Lie algebra is endowed with the negative Jacobi–Lie bracket of vector fields given by n ∂v i ∂wi [v, w]i = L wj − vj j . (1.5.3) j=1 ∂xj ∂x (The sub L on [· , ·] refers to the fact that it is the left Lie algebra bracket on g. The most common convention for the Jacobi–Lie bracket of vector fields, also the one we adopt, has the opposite sign.) We identify g and g∗ using the pairing v, w = v · w d3 x. (1.5.4) Ω Hamiltonian Structure. Introduce the (+) Lie–Poisson bracket, called the ideal fluid bracket, on functions of v by δF δG {F, G}(v) = v· , d3 x, (1.5.5) Ω δv δv L where δF/δv is defined by 1 δF lim [F (v + εδv) − F (v)] = δv · d3 x. (1.5.6) ε→0ε Ω δv With the energy function chosen to be the kinetic energy, 1 2 H(v) = v d3 x, (1.5.7) 2 Ω one can verify that the Euler equations (1.5.2) are equivalent to the Poisson bracket equations ˙ F = {F, H} (1.5.8) ........................... 15 July 1998—18h02 ...........................

35. 1.5 Incompressible Fluids 21 for all functions F on g. For this, one uses the orthogonal decomposition w = Pw + p of a vector field w into a divergence-free part Pw in g and a gradient. The Euler equations can be written ∂v + P(v · v) = 0. (1.5.9) ∂t One can express the Hamiltonian structure in terms of the vorticity as a basic dynamic variable, and show that the preservation of coadjoint orbits amounts to Kelvin’s circulation theorem. Marsden and Weinstein [1983] show that the Hamiltonian structure in terms of Clebsch potentials fits naturally into this Lie–Poisson scheme, and that Kirchhoff’s Hamiltonian description of point vortex dynamics, vortex filaments, and vortex patches can be derived in a natural way from the Hamiltonian structure described above. Lagrangian Structure. The general framework of the Euler-Poincar´ e and the Lie–Poisson equations gives other insights as well. For example, this general theory shows that the Euler equations are derivable from the “variational principle” b 1 2 δ v d3 x = 0 a Ω 2 which is to hold for all variations δv of the form ˙ δv = u + [v, u]L (sometimes called Lin constraints) where u is a vector field (representing the infinitesimal particle displacement) vanishing at the temporal end- points6 . There are important functional analytic differences between working in material representation (that is, on T ∗ G) and in Eulerian representation, that is, on g∗ that are important for proving existence and uniqueness theorems, theorems on the limit of zero viscosity, and the convergence of numerical algorithms (see Ebin and Marsden [1970], Marsden, Ebin, and Fischer [1972], and Chorin, Hughes, Marsden, and McCracken [1978]). Finally, we note that for two-dimensional flow , a collection of Casimir functions is given by C(ω) = Φ(ω(x)) d2 x (1.5.10) Ω for Φ : R → R any (smooth) function where ωk = × v is the vorticity . For three-dimensional flow, (1.5.10) is no longer a Casimir. 6 As mentioned earlier, this form of the variational (strictly speaking a Lagrange d’Alembert type) principle is due to Newcomb [1962]; see also Bretherton [1970]. For the case of general Lie algebras, it is due to Marsden and Scheurle [1993b]; see also Bloch, Krishnaprasad, Marsden and Ratiu [1994b]. See also the review article of Morrison [1994] for a somewhat different perspective. ........................... 15 July 1998—18h02 ...........................

36. 22 1. Introduction and Overview Exercises 1.5-1. Show that any divergence-free vector field X on R3 can be written globally as a curl of another vector field and, away from equilibrium points, can locally be written as X = f × g, where f and g are real-valued functions on R3 . Assume this (so-called Clebsch-Monge) representation also holds globally. Show that the particles of fluid, which follow trajectories satisfying x = X(x), are trajectories of a ˙ Hamiltonian system with a bracket in the form of Exercise 1.3-2. 1.6 The Maxwell–Vlasov System Plasma physics provides another beautiful application area for the techniques discussed in the preceding sections. We shall briefly indicate these in this section. The period 1970–1980 saw the development of noncanonical Hamiltonian structures for the Korteweg-de Vries (KdV) equation (due to Gardner, Kruskal, Miura, and others; see Gardner [1971]) and other soliton equations. This quickly became entangled with the attempts to understand integrability of Hamiltonian systems and the development of the algebraic approach; see, for example, Gelfand and Dorfman [1979], Manin [1979] and references therein. More recently these approaches have come together again; see, for instance, Reyman and Semenov–Tian-Shansky [1990], Moser and Veselov [19–]. KdV type models are usually derived from or are approx- imations to more fundamental fluid models and it seems fair to say that the reasons for their complete integrability are not yet completely understood. Some History. For fluid and plasma systems, some of the key early works on Poisson bracket structures were Dashen and Sharp [1968], Goldin [1971], Iwinski and Turski [1976], Dzyaloshinski and Volovick [1980], Mor- rison and Greene [1980], and Morrison [1980]. In Sudarshan and Mukunda [1974], Guillemin and Sternberg [1982], and Ratiu [1980, 1982], a general theory for Lie–Poisson structures for special kinds of Lie algebras, called semidirect products, was begun. This was quickly recognized (see, for example, Marsden [1982], Marsden, Weinstein, Ratiu, Schmid, and Spencer [1983], Holm and Kuperschmidt [1983], and Marsden, Ratiu and Weinstein [1984a,b]) to be relevant to the brackets for compressible flow; see §1.7 below. Derivation of Poisson Structures. A rational scheme for systematically deriving brackets is needed, since, for one thing, a direct verification of Jacobi’s identity can be inefficient and time-consuming. (See Morrison [1982] and Morrison and Weinstein [1982].) Here we outline a derivation of the Maxwell–Vlasov bracket by Marsden and Weinstein [1982]. The method is similar to Arnold’s, namely by performing a reduction starting with: ........................... 15 July 1998—18h02 ...........................

37. 1.6 The Maxwell–Vlasov System 23 (i) canonical brackets in a material representation for the plasma; and (ii) a potential representation for the electromagnetic field. One then identifies the symmetry group and carries out reduction by this group in a manner similar to that we desribed for Lie–Poisson systems. For plasmas, the physically correct material description is actually slightly more complicated; we refer to Cendra, Holm, Hoyle, and Marsden [1998] for a full account. Parallel developments can be given for many other brackets, such as the charged fluid bracket by Spencer and Kaufman [1982]. Another method, based primarily on Clebsch potentials, was developed in a series of papers by Holm and Kupershmidt (for example, [1983]) and applied to a number of interesting systems, including superfluids and superconductors. They also pointed out that semidirect products were appropriate for the MHD bracket of Morrison and Greene [1980]. The Maxwell–Vlasov System. The Maxwell–Vlasov equations for a collisionless plasma are the fundamental equations in plasma physics7 . In Euclidean space, the basic dynamical variables are: f (x, v, t) : the plasma particle number density per phase space; volume d3 x d3 v; E(x, t) : the electric field; B(x, t) : the magnetic field. The equations for a collisionless plasma for the case of a single species of particles with mass m and charge e are ∂f ∂f e 1 ∂f +v· + E+ v×B · = 0, ∂t ∂x m c ∂v 1 ∂B = −curl E, c ∂t (1.6.1) 1 ∂E 1 = curl B − jf , c ∂t c div E = ρf and div B = 0. The current defined by f is given by jf = e vf (x, v, t) d3 v and the charge density by ρf = e f (x, v, t) d3 v. 7 See, for example, Clemmow and Dougherty [1959], Van Kampen and Felderhof [1967], Krall and Trivelpiece [1973], Davidson [1972], Ichimaru [1973], and Chen [1974]. ........................... 15 July 1998—18h02 ...........................

38. 24 1. Introduction and Overview Also, ∂f /∂x and ∂f /∂v denote the gradients of f with respect to x and v, respectively, and c is the speed of light. The evolution equation for f results from the Lorentz force law and standard transport assumptions. The remaining equations are the standard Maxwell equations with charge density ρf and current jf produced by the plasma. Two limiting cases will aid our discussions. First, if the plasma is constrained to be static, that is, f is concentrated at v = 0 and t-independent, we get the charge-driven Maxwell equations: 1 ∂B = −curl E, c ∂t 1 ∂E (1.6.2) = curl B, c ∂t div E = ρ and div B = 0. Second, if we let c → ∞, electrodynamics becomes electrostatics, and we get the Poisson-Vlasov equation: ∂f ∂f e ∂ϕf ∂f +v· − · = 0, (1.6.3) ∂t ∂x m ∂x ∂v where – 2 ϕf = ρf . In this context, the name “Poisson-Vlasov” seems quite appropriate. The equation is, however, formally the same as the earlier Jeans [1919] equation of stellar dynamics. Hńon [1982] has proposed calling e it the “collisionless Boltzmann equation.” Maxwell’s equations. For simplicity, we let m = e = c = 1. As the basic configuration space, we take the space A of vector potentials A on R3 (for the Yang–Mills equations this is generalized to the space of connections on a principal bundle over space). The corresponding phase space T ∗ A is identified with the set of pairs (A, Y), where Y is also a vector field on R3 . The canonical Poisson bracket is used on T ∗ A : δF δG δF δG {F, G} = − d3 x. (1.6.4) δA δY δY δA The electric field is E = −Y and the magnetic field is B = curl A. With the Hamiltonian 1 2 H(A, Y) = ( E + B 2 ) d3 x, (1.6.5) 2 Hamilton’s canonical field equations (1.1.14) are checked to give the equations for ∂E/∂t and ∂A/∂t which imply the vacuum Maxwell’s equations. Alternatively, one can begin with T A and the Lagrangian ˙ 1 ˙ L(A, A) = A 2 − ×A 2 d3 x (1.6.6) 2 ........................... 15 July 1998—18h02 ...........................

39. 1.6 The Maxwell–Vlasov System 25 and use the Euler–Lagrange equations and variational principles. It is of interest to incorporate the equation div E = ρ and, correspondingly, to use directly the field strengths E and B, rather than E and A. To do this, we introduce the gauge group G, the additive group of real-valued functions ψ : R3 → R. Each ψ ∈ G transforms the fields according to the rule (A, E) → (A + ψ, E). (1.6.7) Each such transformation leaves the Hamiltonian H invariant and is a canonical transformation, that is, it leaves Poisson brackets intact. In this situation, as above, there will be a corresponding conserved quantity, or momentum map in the same sense as in §1.3. As mentioned there, some simple general formulas for computing them will be studied in detail in Chapter 12. For the action (1.6.7) of G on T ∗ A, the associated momentum map is J(A, Y) = div E, (1.6.8) so we recover the fact that div E is preserved by Maxwell’s equations (this is easy to verify directly using div curl = 0). Thus we see that we can incorporate the equation div E = ρ by restricting our attention to the set J−1 (ρ). The theory of reduction is a general process whereby one reduces the dimension of a phase space by exploiting conserved quantities and symmetry groups. In the present case, the reduced space is J−1 (ρ)/G which is identified with Maxρ , the space of E’s and B’s satisfying div E = ρ and div B = 0. The space Maxρ inherits a Poisson structure as follows. If F and K are functions on Maxρ , we substitute E = −Y and B = × A to express F and K as functionals of (A, Y). Then we compute the canonical brackets on T ∗ A and express the result in terms of E and B. Carrying this out using the chain rule gives δF δK δK δF {F, K} = · curl − · curl d3 x, (1.6.9) δE δB δE δB where δF/δE and δF/δB are vector fields, with δF/δB divergence-free. These are defined in the usual way; for example, 1 δF lim [F (E + εδE, B) − F (E, B)] = · δE d3 x. (1.6.10) ε→0ε δE This bracket makes Maxρ into a Poisson manifold and the map (A, Y) → (−Y, × A) into a Poisson map. The bracket (1.6.9) was discovered (by a different procedure) by Pauli [1933] and Born and Infeld [1935]. We refer to (1.6.9) as the Pauli-Born-Infeld bracket or the Maxwell–Poisson bracket for Maxwell’s equations. ........................... 15 July 1998—18h02 ...........................

40. 26 1. Introduction and Overview With the energy H given by (1.6.5) regarded as a function of E and B, ˙ Hamilton’s equations in bracket form F = {F, H} on Maxρ captures the full set of Maxwell’s equations (with external charge density ρ). The Poisson-Vlasov Equation. Morrison [1980] showed that the Poisson- Vlasov equations form a Hamiltonian system with 1 1 H(f ) = v 2 f (x, v, t) d3 x d3 v + ϕf 2 d3 x (1.6.11) 2 2 and the Poisson-Vlasov bracket δF δG {F, G} = f , d3 x d3 v, (1.6.12) δf δf xv where { , }xv is the canonical bracket on (x, v)-space. As was observed in Gibbons [1981] and Marsden and Weinstein [1982], this is the (+) Lie– Poisson bracket associated with the Lie algebra g of functions of (x, v) with Lie bracket the canonical Poisson bracket. According to the general theory, this Lie–Poisson structure is obtained by reduction from canonical brackets on the cotangent bundle of the group underlying g, just as was the case for the rigid body and incompressible fluids. This time the group G = Diff can is the group of canonical transformations of (x, v)-space. The Poisson-Vlasov equations can equally well be written in canonical form on T ∗ G. This is the Lagrangian description of a plasma, and the Hamiltonian description here goes back to Low [1958], Katz [1961], and Lundgren [1963]. Thus, one can start with the Lagrangian description with canonical brackets and, through reduction, derive the brackets here. There are other approaches to the Hamiltonian formulation using analogs of Clebsch potentials; see, for instance, Su [1961], Zakharov [1971], and Gibbons, Holm, and Kupershmidt [1982]. See Cendra, Holm, Hoyle, and Marsden [1998] for further information on these topics. The Poisson-Vlaslov to Compressible Flow Map. Before going on to the Maxwell–Vlasov equations, we point out a remarkable connection between the Poisson-Vlasov bracket (1.6.12) and the bracket for compressible flow. The Euler equations for compressible flow in a region Ω in R3 are ∂v ρ + (v · )v =− p (1.6.13) ∂t and ∂ρ + div(ρv) = 0, (1.6.14) ∂t with the boundary condition v tangent to ∂Ω. ........................... 15 July 1998—18h02 ...........................

41. 1.6 The Maxwell–Vlasov System 27 Here the pressure p is determined from an internal energy function per unit mass given by p = ρ2 w (ρ), where w = w(ρ) is the constitutive relation. (We ignore entropy for the present discussion—its inclusion is starightfor- ward to deal with.) The compressible fluid Hamiltonian is 1 2 H= ρ v d3 x + ρw(ρ) d3 x. (1.6.15) 2 Ω Ω The relevant Poisson bracket is most easily expressed if we use the momentum density M = ρv and density ρ as our basic variables. The compressible fluid bracket is δG δF δF δG 3 {F, G} = M· · − · d x Ω δM δM δM δM δG δF δF δG 3 + ρ · − · d x. (1.6.16) Ω δM δρ δM δρ The space of (M, ρ)’s can be shown to be the dual of a semidirect product Lie algebra and that the preceding bracket is the associated (+) Lie–Poisson bracket (see Marsden, Weinstein, Ratiu, Schmid, and Spencer [1983], Holm and Kupershmidt [1983], and Marsden, Ratiu, and Weinstein [1984a,b]). The relationship with the Poisson-Vlasov bracket is this: suppressing the time variable, define the map f → (M, ρ) by M(x) = vf (x, v)d3 v and ρ(x) = f (x, v) d3 v. (1.6.17) Ω Ω Remarkably, this plasma to fluid map is a Poisson map taking the Poisson- Vlasov bracket (1.6.12) to the compressible fluid bracket (1.6.16). In fact, this map is a momentum map (Marsden, Weinstein, Ratiu, Schmid, and Spencer [1983]). The Poisson-Vlasov Hamiltonian is not invariant under the associated group action, however. The Maxwell–Vlasov Bracket. A bracket for the Maxwell–Vlasov equations was given by Iwinski and Turski [1976] and Morrison [1980]. Marsden and Weinstein [1982] used systematic procedures involving reduction and momentum maps to derive (and correct) the bracket from a canonical bracket. The procedure starts with the material description of the plasma as the cotangent bundle of the group Diff can of canonical transformations of (x, p)-space and the space T ∗ A for Maxwell’s equations. We justify this by noticing that the motion of a charged particle in a fixed, but (possibly time-dependent) electromagnetic field via the Lorentz force law defines a (time-dependent) canonical transformation. On T ∗ Diff can ×T ∗ A we put the sum of the two canonical brackets, and then we reduce. First we reduce by Diff can , which acts on T ∗ Diff can by right translation, but does not act on ........................... 15 July 1998—18h02 ...........................

42. 28 1. Introduction and Overview T ∗ A. Thus we end up with densities fmom (x, p, t) on position-momentum space and with the space T ∗ A used for the Maxwell equations. On this space we get the (+) Lie–Poisson bracket, plus the canonical bracket on T ∗ A. Recalling that p is related to v and A by p = v + A, we let the gauge group G of electromagnetism act on this space by (fmom (x, p, t), A(x, t), Y(x, t)) → (fmom (x, p + ϕ(x), t), A(x, t) + ϕ(x), Y(x, t)). (1.6.18) The momentum map associated with this action is computed to be J(fmom , A,Y) = div E − fmom (x, p) d3 p. (1.6.19) This corresponds to div E − ρf if we write f (x, v, t) = fmom (x, p − A, t). This reduced space J−1 (0)/G can be identiﬁed with the space MV of triples (f, E, B), satisfying div E = ρf and div B = 0. The bracket on MV is computed by the same procedure as for Maxwell’s equations. These computations yield the following Maxwell–Vlasov bracket: δF δK {F, K}(f, E, B) = f , d3 x d3 v δf δf xv δF δK δK δF + · curl − · curl d3 x δE δB δE δB (1.6.20) δF δf δK δK δf δF + · − · 3 d xd v 3 δE δv δf δE δv δf ∂ δF ∂ δK + fB · × d3 x d3 v. ∂v δf ∂v δf With the Maxwell–Vlasov Hamiltonian 1 H(f, E, B) = v 2 f (x, v, t) d3 x d3 v 2 1 2 + ( E(x, t) + B(x, t) 2 ) d3 x, 2 the Maxwell–Vlasov equations take the Hamiltonian form ˙ F = {F, H} (1.6.21) on the Poisson manifold MV. Exercises 1.6-1. Verify that one obtains the Maxwell equations from the Maxwell– Poisson bracket. 1.6-2. Verify that the action (1.6.7) has the momentum map J(A, Y) = div E in the sense given in §1.3. ........................... 15 July 1998—18h02 ...........................

43. 1.7 Nonlinear Stability 29 1.7 Nonlinear Stability There are various meanings that can be given to the word “stability.” In- tuitively, stability means that small disturbances do not grow large as time passes. Being more precise about this notion is not just mathematical nit- picking; indeed, different interpretations of the word stability can lead to different stability criteria. Examples like the double spherical pendulum and stratified shear flows that are sometimes used to model oceanographic phenomena, show that one can get different criteria if one uses linearized or nonlinear analyses (see Marsden and Scheurle [1993a] and Abarbanel, Holm, Marsden, and Ratiu [1986]). Some History. The history of stability theory in mechanics is very complex, but certainly has its roots in the work of Riemann [1860, 1861], Routh [1877], Thomson and Tait [1879], Poincar´ [1885, 1892], and Lia- e punov [1892, 1897]. Since these early references, the literature has become too vast to even survey roughly. We do mention however, that a guide to the large Soviet literature may be found in Mikhailov and Parton [1990]. The basis of the nonlinear stability method discussed below was originally given by Arnold [1965b, 1966b] and applied to two-dimensional ideal fluid flow, substantially augmenting the pioneering work of Lord Rayleigh [1880]. Related methods were also found in the plasma physics literature, notably by Newcomb [1958], Fowler [1963], and Rosenbluth [1964]. However, these works did not provide a general setting or key convexity estimates needed to deal with the nonlinear nature of the problem. In retrospect, we may view other stability results, such as the stability of solitons in the Korteweg-de Vries (KdV) equations due to Benjamin [1972] and Bona [1975] (see also Maddocks and Sachs [1992]) as being instances of the same method used by Arnold. A crucial part of the method exploits the fact that the basic equations of nondissipative fluid and plasma dynamics are Hamiltonian in character. We shall explain below how the Hamiltonian structures discussed in the previous sections are used in the stability analysis. Dynamics and Stability. Stability is a dynamical concept. To explain it, we shall use some fundamental notions from the theory of dynamical systems (see, for example, Hirsch and Smale [1974] and Guckenheimer and Holmes [1983]). The laws of dynamics are usually presented as equations of motion which we write in the abstract form of a dynamical system: u = X(u). ˙ (1.7.1) Here, u is a variable describing the state of the system under study, X is a system-specific function of u and u = du/dt, where t is time. The set of ˙ all allowed u’s forms the phase space P . For a classical mechanical system, u is often a 2n-tuple (q 1 , . . . , q n , p1 , . . . , pn ) of positions and momenta and, ........................... 15 July 1998—18h02 ...........................

44. 30 1. Introduction and Overview for fluids, u is a velocity field in physical space. As time evolves, the state of the system changes; the state follows a curve u(t) in P . The trajectory u(t) is assumed to be uniquely determined if its initial condition u0 = u(0) is specified. An equilibrium state is a state ue such that X(ue ) = 0. The unique trajectory starting at ue is ue itself; that is, ue does not move in time. The language of dynamics has been an extraordinarily useful tool in the physical and biological sciences, especially during the last few decades. The study of systems which develop spontaneous oscillations through a mechanism called the Poincar´-Andronov-Hopf bifurcation is an example of such e a tool (see Marsden and McCracken [1976], Carr [1981], and Chow and Hale [1982], for example). More recently, the concept of “chaotic dynamics” has sparked a resurgence of interest in dynamical systems. This occurs when dynamical systems possess trajectories that are so complex that they behave as if they were random. Some believe that the theory of turbulence will use such notions in its future development. We are not concerned with chaos directly, although it plays a role in some of what follows. In particular, we remark that in the definition of stability below, stability does not preclude chaos. In other words, the trajectories near a stable point can still be temporally very complex; stability just prevents them from moving very far from equilibrium. To define stability, we choose a measure of nearness in P using a “metric” d. For two points u1 and u2 in P , d determines a positive number denoted d(u1 , u2 ), which is called the distance from u1 to u2 . In the course of a stability analysis, it is necessary to specify, or construct, a metric appropriate for the problem at hand. In this setting, one says that an equilibrium state ue is stable when trajectories which start near ue remain near ue for all t ≥ 0. In precise terms, given any number > 0, there is δ > 0 such that if d(u0 , ue ) < δ, then d(u(t), ue ) < for all t > 0 . Figure 1.7.1 shows examples of stable and unstable equilibria for dynamical systems whose state space is the plane. Fluids can be stable relative to one distance measure and, simultaneously, unstable relative to another. This seeming pathology actually reflects important physical processes; see Wan and Pulvirente [1984]. Rigid Body Stability. A physical example illustrating the definition of stability is the motion of a free rigid body. This system can be simulated by tossing a book, held shut with a rubber band, into the air. It rotates stably when spun about its longest and shortest axes, but unstably when spun about the middle axis (Figure 1.7.2). The distance measure defining stability in this example is a metric in body angular momentum space. We shall return to this example in detail in Chapter 15 when we study rigid body stability. Linearized and Spectral Stability. There are two other ways of treating stability. First of all, one can linearize equation (1.7.1); if δu denotes a ........................... 15 July 1998—18h02 ...........................

45. 1.7 Nonlinear Stability 31 ue ue ue ue (a) (b) (c) (d) Figure 1.7.1. The equilibrium point (a) is unstable because the trajectory u(t) does not remain near ue . Similarly (b) is unstable since most trajectories (eventually) move away from ue . The equilibria in (c) and (d) are stable because all trajectories near ue stay near ue . (a) (b) (c) Figure 1.7.2. If you toss a book into the air, you can make it spin stably about its shortest axis (a), and its longest axis (b), but it is unstable when it rotates about its middle axis (c). variation in u and X (ue ) denotes the linearization of X at ue (the matrix of partial derivatives in the case of finitely many degrees of freedom), the linearized equations describe the time evolution of “infinitesimal” disturbances of ue : d (δu) = X (ue ) · δu. (1.7.2) dt Equation (1.7.1), on the other hand, describes the nonlinear evolution of finite disturbances ∆u = u − ue . We say ue is linearly stable if (1.7.2) is stable at δu = 0, in the sense defined above. Intuitively, this means that there are no infinitesimal disturbances which are growing in time. If (δu)0 ........................... 15 July 1998—18h02 ...........................

46. 32 1. Introduction and Overview is an eigenfunction of X (ue ), that is, if X (ue ) · (δu)0 = λ(δu)0 (1.7.3) for a complex number λ, then the corresponding solution of (1.7.2) with initial condition (δu)0 is δu = etλ (δu)0 . (1.7.4) This is growing when λ has positive real part. This leads us to the third notion of stability: we say that (1.7.1) or (1.7.2) is spectrally stable if the eigenvalues (more precisely points in the spectrum) all have non-positive real parts. In finite dimensions and, under appropriate technical conditions in infinite dimensions, one has the following implications: (stability) ⇒ (spectral stability) and (linear stability) ⇒ (spectral stability). If the eigenvalues all lie strictly in the left half-plane, then a classical result of Liapunov guarantees stability. (See, for instance, Hirsch and Smale [1974] for the finite-dimensional case and Marsden and McCracken [1976], or Abraham, Marsden, and Ratiu [1988] for the infinite-dimensional case.) However, in systems of interest to us, the dissipation is very small; our systems will often be conservative. For such systems the eigenvalues must be symmetrically distributed under reflection in the real and imaginary axis. This implies that the only possibility for spectral stability is when the eigenvalues lie exactly on the imaginary axis. Thus, this version of the Liapunov theorem is of no help in the Hamiltonian case. Spectral stability need not imply stability; instabilities can be generated (even in Hamiltonian systems) through, for example, resonance. Thus, to obtain general stability results, one must use other techniques to augment or replace the linearized theory. We give such a technique below. Here is a planar example of a system which is spectrally stable at the origin, but which is unstable there. In polar coordinates (r, θ), consider the evolution of u = (r, θ) given by r = r3 (1 − r2 ) ˙ ˙ and θ = 1. (1.7.5) In (x, y) coordinates this system takes the form x = x(x2 + y 2 )(1 − x2 − y 2 ) − y, y = y(x2 + y 2 )(1 − x2 − y 2 ) + x. ˙ ˙ The eigenvalues of the linearized system at the origin are readily verified √ to be ± −1, so the origin is spectrally stable; however, the phase portrait, shown in Figure 1.7.3 shows that the origin is unstable. (We include the factor 1 − r2 to give the system an attractive periodic orbit—this is merely ........................... 15 July 1998—18h02 ...........................

47. 1.7 Nonlinear Stability 33 to enrich the example and show how a stable periodic orbit can attract the orbits expelled by an unstable equilibrium.) This is not, however, a conservative system; next we give two examples of Hamiltonian systems with similar features. ˙ Figure 1.7.3. The phase portrait for r = r3 (1 − r2 ); θ = 1. ˙ Resonance Example. The linear system in R2 whose Hamiltonian is given by 1 2 1 2 H(q, p) = p + q + pq 2 2 has zero as a double eigenvalue so it is spectrally stable. On the other hand, q(t) = (q0 + p0 )t + q0 and p(t) = −(q0 + p0 )t + p0 is the solution of this system with initial condition (q0 , p0 ), which clearly leaves any neighborhood of the origin no matter how close to it (q0 , p0 ) is. Thus spectral stability need not imply even linear stability. An even simpler example of the same phenomenon is given by the free particle Hamiltonian H(q, p) = 1 p2 . 2 Another higher-dimensional example with resonance in R8 is given by the linear system whose Hamiltonian is H = q 2 p1 − q 1 p 2 + q 4 p 3 − q 3 p4 + q 2 q 3 . The general solution with initial condition (q1 , . . . , p0 ) is given by 0 4 0 0 q1 (t) = q1 cos t + q2 sin t, q2 (t) = −q1 sin t + q2 cos t, 0 0 0 0 q3 (t) = q3 cos t + q4 sin t, q4 (t) = −q3 sin t + q4 cos t, 0 0 ........................... 15 July 1998—18h02 ...........................

48. 34 1. Introduction and Overview 0 q3 q0 p1 (t) = − t sin t + 4 (t cos t − sin t) + p0 cos t + p0 sin t, 1 2 2 2 0 0 q q p2 (t) = − 3 (t cos t + sin t) − 4 t sin t − p0 sin t + p0 cos t, 1 2 2 2 0 0 q q p3 (t) = 1 t sin t − 2 (t cos t + sin t) + p0 cos t + p0 sin t, 3 4 2 2 0 0 q q p4 (t) = 1 (t cos t − sin t) + 2 t sin t − p0 sin t + p0 cos t. 3 4 2 2 One sees that pi (t) leaves any neighborhood of the origin, no matter how close to the origin the initial conditions (q1 , . . . , p0 ) are, that is, the system 0 4 is linearly unstable. On the other hand, all eigenvalues of this linear system are ±i, each a quadruple eigenvalue. Thus, this linear system is spectrally stable. Cherry’s Example (Cherry [1959,1968]). This example is a Hamil- tonian system that is spectrally stable and linearly stable but is nonlinearly unstable. Consider the Hamiltonian on R4 given by 1 2 1 H= (q + p2 ) − (q2 + p2 ) + p2 (p2 − q1 ) − q1 q2 p1 . 2 2 (1.7.6) 2 1 1 2 2 1 This system has an equilibrium at the origin, which is linearly stable since the linearized system consists of two uncoupled oscillators in the (δq2 , δp2 ) and (δq1 , δp1 ) variables, respectively, with frequencies in the ratio 2 : 1 (the eigenvalues are ±i and ±2i, so the frequencies are in resonance). A family of solutions (parametrized by a constant τ ) of Hamilton’s equations for (1.7.6) is given by  √ cos(t − τ ) cos 2(t − τ )  q1 = − 2 , q2 = ,    t−τ t−τ (1.7.7) √ sin(t − τ ) sin 2(t − τ )   p1 = 2 , p2 = .  t−τ t−τ The solutions (1.7.7) clearly blow up in finite time; however, they start at √ time t = 0 at a distance 3/τ from the origin, so by choosing τ large, we can find solutions starting arbitrarily close to the origin, yet going to infinity in a finite time, so the origin is nonlinearly unstable. Despite the above situation relating the linear and nonlinear theories, there has been much effort devoted to the development of spectral stability methods. When instabilities are present, spectral estimates give important information on growth rates. As far as stability goes, spectral stability gives necessary, but not sufficient, conditions for stability. In other words, for the nonlinear problems spectral instability can predict instability, but not stability, this is a basic result of Liapunov; see Abraham, Marsden, and Ratiu [1988], for example. Our immediate purpose is the opposite: to describe sufficient conditions for stability. ........................... 15 July 1998—18h02 ...........................

49. 1.7 Nonlinear Stability 35 Casimir Functions. Besides the energy, there are other conserved quantities associated with group symmetries such as linear and angular momentum. Some of these are associated with the group that underlies the passages from material to spatial or body coordinates. These are called Casimir functions; such a quantity, denoted C, is characterized by the fact that it Poisson commutes with every function, that is {C, F } = 0 (1.7.8) for all functions F on phase space P . We shall study such functions and their relation with momentum maps in Chapters 10 and 11. For example, if Φ is any function of one variable, the quantity C(Π) = Φ( Π 2 ) (1.7.9) is a Casimir for the rigid body bracket, as is seen by using the chain rule. Likewise, C(ω) = Φ(ω) dx dy (1.7.10) Ω is a Casimir function for the two-dimensional ideal fluid bracket. (This calculation ignores boundary terms that arise in an integration by parts— see Lewis, Marsden, Montgomery, and Ratiu [1986] for a treatment of these boundary terms.) Casimir functions are conserved by the dynamics associated with any ˙ Hamiltonian H since C = {C, H} = 0. Conservation of (1.7.9) corresponds to conservation of total angular momentum for the rigid body, while conservation of (1.7.10) represents Kelvin’s circulation theorem for the Euler equations. It provides infinitely many independent constants of the motion that mutually Poisson commute; that is, {C1 , C2 } = 0, but this does not imply that these equations are integrable. Lagrange–Dirichlet Criterion. For Hamiltonian systems in canonical form, an equilibrium point (qe , pe ) is a point at which the partial derivatives of H vanish, that is, it is a critical point of H. If the 2n × 2n matrix δ 2 H f second partial derivatives evaluated at (qe , pe ) is positive- or negative- definite (that is, all the eigenvalues of δ 2 H(qe , pe ) have the same sign), then (qe , pe ) is stable. This follows from conservation of energy and the fact from calculus, that the level sets of H near (qe , pe ) are approximately ellipsoids. As mentioned earlier, this condition implies, but is not implied by, spectral stability. The KAM (Kolmogorov, Arnold, Moser) theorem, which gives stability of periodic solutions for two degree of freedom systems, and the Lagrange–Dirichlet theorem are the most basic general stability theorems for equilibria of Hamiltonian systems. For example, let us apply the Lagrange–Dirichlet theorem to a classical mechanical system whose Hamiltonian is the form kinetic plus potential ........................... 15 July 1998—18h02 ...........................

50. 36 1. Introduction and Overview energy. If (qe , pe ) is an equilibrium, it follows that pe is zero. Moreover, the matrix δ 2 H of second-order partial derivatives of H evaluated at (qe , pe ) block diagonalizes with one of the blocks being the matrix of the quadratic form of the kinetic energy which is always positive-definite. Therefore, if δ 2 H is definite, it must be positive-definite and this in turn happens if and only if δ 2 V is positive-definite at qe , where V is the potential energy of the system. We conclude that for a mechanical system whose Lagrangian is kinetic minus potential energy, (qe , 0) is a stable equilibrium, provided the matrix δ 2 V (qe ) of second-order partial derivatives of the potential V at qe is positive-definite (or, more generally, qe is a strict local minimum for V ). If δ 2 V at qe has a negative-definite direction, then qe is an unstable equilibrium. The second statement is seen in the following way. The linearized Hamil- tonian system at (qe , 0) is again a Hamiltonian system whose Hamiltonian is of the form kinetic plus potential energy, the potential energy being given by the quadratic form δ 2 V (qe ). From a standard theorem in linear algebra, which states that two quadratic forms, one of which is positive-definite, can be simultaneously diagonalized, we conclude that the linearized Hamilto- nian system decouples into a family of Hamiltonian systems of the form d d 1 (δpk ) = −ck δq k , (δq k ) = δpk , dt dt mk where 1/mk > 0 are the eigenvalues of the positive-definite quadratic form given by the kinetic energy in the variables δpj , and ck are the eigenvalues of δ 2 V (qe ). Thus the eigenvalues of the linearized system are given by ± −ck /mk . Therefore, if some ck is negative, the linearized system has at least one positive eigenvalue and thus (qe , 0) is spectrally and hence linearly and nonlinearly unstable. For generalizations of this, see Oh [1987], Strauss [1987], Chern [1997] and references therein. The Energy-Casimir Method. This is a generalization of the classical Lagrange–Dirichlet method. Given an equilibrium ue for u = XH (u) on a ˙ Poisson manifold P , it proceeds in the following steps. To test an equilibrium (satisfying XH (ze ) = 0) for stability: Step 1. Find a conserved function C (C will typically be a Casimir function plus other conserved quantities) such that the first variation vanishes: δ(H + C)(ze ) = 0. Step 2. Calculate the second variation δ 2 (H + C)(ze ). Step 3. If δ 2 (H + C)(ze ) is definite (either positive or negative), then ze is called formally stable. ........................... 15 July 1998—18h02 ...........................

51. 1.7 Nonlinear Stability 37 With regard to Step 3, we point out that an equilibrium solution need not be a critical point of H alone; in general, δH(ze ) = 0. An example where this occurs is a rigid body spinning about one of its principal axes of inertia. In this case, a critical point of H alone would have zero angular velocity; but a critical point of H + C is a (nontrivial) stationary rotation about one of the principal axes. The argument used to establish the Lagrange–Dirichlet test formally works in infinite dimensions too. Unfortunately, for systems with infinitely many degrees of freedom (like fluids and plasmas), there is a serious technical snag. The calculus argument used before runs into problems; one might think these are just technical and that we just need to be more careful with the calculus arguments. In fact, there is widespread belief in this “energy criterion” (see, for instance, the discussion and references in Marsden and Hughes [1983], Chapter 6, and Potier–Ferry [1982]). However, Ball and Marsden [1984] have shown using an example from elasticity theory that the difficulty is genuine: they produce a critical point of H at which δ 2 H is positive-definite, yet this point is not a local minimum of H. On the other hand, Potier–Ferry [1982] shows that asymptotic stability is restored if suitable dissipation is added. Another way to overcome this difficulty is to modify Step 3 using a convexity argument of Arnold [1966b]. Modified Step 3. Assume P is a linear space. (a) Let ∆u = u − ue denote a finite variation in phase space . (b) Find quadratic functions Q1 and Q2 such that Q1 (∆u) ≤ H(ue + ∆u) − H(ue ) − δH(ue ) · ∆u and Q2 (∆u) ≤ C(ue + ∆u) − C(ue ) − δC(ue ) · ∆u, (c) Require that Q1 (∆u) + Q2 (∆u) > 0 for all ∆u = 0. (d) Introduce the norm ∆u by 2 ∆u = Q1 (∆u) + Q2 (∆u), so ∆u is a measure of the distance from u to ue : d(u, ue ) = ∆u . (e) Require that |H(ue + ∆u) − H(ue )| ≤ C1 ∆u α and |C(ue + ∆u) − C(ue )| ≤ C2 ∆u α for constants α, C1 , C2 > 0, and ∆u sufficiently small. ........................... 15 July 1998—18h02 ...........................

52. 38 1. Introduction and Overview These conditions guarantee stability of ue and provide the distance measure relative to which stability is defined. The key part of the proof is simply the observation that if we add the two inequalities in (b), we get ∆u 2 ≤ H(ue + ∆u) + C(ue + ∆u) − H(ue ) − C(ue ) using the fact that δH(ue ) · ∆u and δC(ue ) · ∆u add up to zero by Step 1. But H and C are constant in time so (∆u)time=t 2 ≤ [H(ue + ∆u) + C(ue + ∆u) − H(ue ) − C(ue )]|time=0 . Now employ the inequalities in (e) to get (∆u)time=t 2 ≤ (C1 + C2 ) (∆u)time=0 α . This estimate bounds the temporal growth of finite perturbations in terms of initial perturbations, which is what is needed for stability. For a survey of this method, additional references and numerous examples, see Holm, Marsden, Ratiu, and Weinstein [1985]. There are some situations (such as the stability of elastic rods) in which the above techniques do not apply. The chief reason is that there may be a lack of sufficiently many Casimir functions to even achieve the first step. For this reason a modified (but more sophisticated) method has been developed called the “energy-momentum method.” The key to the method is to avoid the use of Casimir functions by applying the method before any reduction has taken place. This method was developed in a series of papers of Simo, Posbergh, and Marsden [1990, 1991] and Simo, Lewis, and Marsden [1991]. A discussion and additional references are found later in this section. Gyroscopic Systems. The distinctions between “stability by energy methods, that is, energetics” and “spectral stability,” become especially interesting when one adds dissipation. In fact, building on the classical work of Kelvin and Chetaev, one can prove that if δ 2 H is indefinite, yet the spectrum is on the imaginary axis, then adding dissipation necessarily makes the system linearly unstable. That is, at least one pair of eigenvalues of the linearized equations move into the right half-plane. This is a phenomenon called dissipation induced instability. This result, along with related developments, is proved in Bloch, Krishnaprasad, Marsden, and Ratiu [1991, 1994, 1996]. For example, consider the linear gyroscopic system ¨ ˙ M q + S q + V q = 0, (1.7.11) where q ∈ Rn , M is a positive-definite symmetric n × n matrix, S is skew, and V is symmetric. This system is Hamiltonian (Exercise 1.7-2). If V has negative eigenvalues, then (1.7.11) is formally unstable. However, due to ........................... 15 July 1998—18h02 ...........................

53. 1.7 Nonlinear Stability 39 S, the system can be spectrally stable. However, if R is positive-definite symmetric and > 0 is small, the system with friction ¨ ˙ ˙ M q + S q + Rq + V q = 0 (1.7.12) is linearly unstable. A specific example is given in Exercise 1.7-4. Outline of the energy-momentum method. The energy momentum method is an extension of the Arnold (or energy-Casimir) method for the study of stability of relative equilibria, which was developed for Lie–Poisson systems on duals of Lie algebras, especially those of fluid dynamical type. In addition, the method extends and refines the fundamental stability techniques going back to Routh, Liapunov and in more recent times, to the work of Smale. The motivation for these extensions is three fold. First of all, the energy-momentum method can deal with Lie–Poisson systems for which there are not sufficient Casimir functions available, such as 3D ideal flow and certain problems in elasticity. In fact, Abarbanel and Holm [1987] use what can be recognized retrospectively is the energy- momentum method to show that 3d equilibria for ideal flow are always formally unstable due to vortex stretching. Other fluid and plasma situations, such as those considered by Chern and Marsden [1990] for ABC flows, and certain multiple hump situations in plasma dynamics (see Holm, Marsden, Ratiu and Weinstein [1985] and Morrison [1987] for example) provided additional motivation in the Lie–Poisson setting. A second motivation is to extend the method to systems that need not be Lie–Poisson and still make use of the powerful idea of using reduced spaces, as in the original Arnold method. Examples such as rigid bodies with vi- brating antennas (Sreenath, et al [1988], Oh et al [1989], Krishnaprasad and Marsden [1987]) and coupled rigid bodies (Patrick [1989]) motivated the need for such an extension of the theory. Finally, it gives sharper stability conclusions in material representation and links with geometric phases. The idea of the energy-momentum method. The setting of the energy-momentum method is that of a mechanical system with symmetry with a configuration space Q and phase space T ∗ Q and a symmetry group G acting, with a standard momentum map J : T ∗ Q → g∗ , where g∗ is the Lie algebra of G. Of course one gets the Lie–Poisson case when Q = G. The rough idea for the energy momentum method is to first formulate the problem directly on the unreduced space. Here, relative equilibria associated with a Lie algebra element ξ are always critical points of the augmented Hamiltonian Hξ := H − J, ξ . The idea is to now compute the second variation of Hξ at a relative equilibria ze with momentum value µe subject to the constraint J = µe and on a space transverse to the action of Gµe . Although the augmented Hamiltonian plays the role of H + C in ........................... 15 July 1998—18h02 ...........................

54. 40 1. Introduction and Overview the Arnold method, notice that Casimir functions are not required to carry out the calculations. The surprising thing is that the second variation of Hξ at the relative equilibrium can be arranged to be block diagonal, using splittings that are based on the mechanical connection while, at the same time, the symplectic structure also has a simple block structure so that the linearized equations are put into a useful canonical form. Even in the Lie–Poisson setting, this leads to situations in which one gets much simpler second variations. This block diagonal structure is what gives the method its computational power. The general theory for carrying out this procedure was developed in Simo, Posbergh and Marsden [1990, 1991] and Simo, Lewis and Marsden [1991]. An exposition of the method may be found, along with additional references in Marsden [1992]. It has been extended to the singular case by Ortega and Ratiu [1997b]. Lagrangian version of the energy-momentum method. The energy momentum method may also be usefully formulated in the Lagrangian setting and this setting is very convenient for the calculations in many examples. The general theory for this was done in Lewis [1992] and Wang and Krishnaprasad [1992]. This Lagrangian setting is closely related to the general theory of Lagrangian reduction we shall come to later on. In this context one reduces variational principles rather than symplectic and Poisson structures and for the case of reducing the tangent bundle of a Lie group, it leads to the Euler-Poincar´ equations rather than the Lie–Poisson e equations. Effectiveness in examples. The energy momentum method has proven its effectiveness in a number of examples. For instance, Lewis and Simo [1990] were able to deal with the stability problem for pseudo-rigid bodies, which was thought up to that time to be analytically intractable. The energy-momentum method can sometimes be used in contexts where the reduced space is singular or at nongeneric points in the dual of the Lie algebra. This is done at singular points in Lewis, Ratiu, Simo and Marsden [1992] who analyze the heavy top in great detail and, in the Lie– Poisson setting for compact groups at nongeneric points in the dual of the Lie algebra, in Patrick [1992, 1995]. One of the key things is to keep track of group drifts because the isotropy group Gµ can change for nearby points, and these are of course very important for the reconstruction process and for understanding the Hannay-Berry phase in the context of reduction (see Marsden, Ratiu and Montgomery [1990] and references therein). For noncompact groups and an application to the dynamics of rigid bodies in fluids (underwater vehicles), see Leonard and Marsden [1997]. Additional work in this area is still needed in the context of singular reduction. The celebrated Benjamin–Bona theorem on stability of solitons for the KdV equation can be viewed as an instance of the energy momentum method, see also Maddocks and Sachs [195?], and for example, Oh [1987] ........................... 15 July 1998—18h02 ...........................

55. 1.7 Nonlinear Stability 41 and Grillakis Shatah and Strauss [1987], although of course there are many subtelties in the pde context. Hamiltonian bifurcations. The energy-momentum method has also been used in the context of Hamiltonian bifurcation problems. One such context is that of free boundary problems building on the work of Lewis, Montgomery, Marsden and Ratiu [1986] which gives a Hamiltonian structure for dynamic free boundary problems (surface waves, liquid drops, etc), generalizing Hamiltonian structures found by Zakharov. Along with the Arnold method itself, this is used for a study of the bifurcations of such problems in Lewis, Marsden and Ratiu [1987], Lewis, [1989, 1992], Kruse, Marsden, and Scheurle [1993] and other references cited therein. Converse to the energy-momentum method. Because of the block structure mentioned, it has also been possible to prove, in a sense, a converse of the energy-momentum method. That is, if the second variation is indefinite, then the system is unstable. One cannot, of course hope to do this literally as stated since there are many systems (eg, examples studied by Chetayev) which are formally unstable, and yet their linearizations have eigenvalues lying on the imaginary axis. Most of these are presumably unstable due to Arnold diffusion, but of course this is a very delicate situation to prove analytically. Instead, the technique is to show that with the addition of dissipation, the system is destabilized. This idea of dissipation induced instability goes back to Thomson and Tait in the last century. In the context of the energy-momentum method, Bloch, Krishnaprasad, Marsden and Ratiu [1994,1996] show that with the addition of appropriate dissipation, the indefinitness of the second variation is sufficient to induce linear instability in the problem. There are related eigenvalue movement formulas (going back to Krein) that are used to study non-Hamiltonian perturbations of Hamiltonian normal forms in Kirk, Marsden and Silber [1996]. There are interesting analogs of this for reversible systems in O’Reilly, Malhotra, and Namamchchivaya [1996]. Extension of the energy-momentum method to nonholonomic systems. The energy-momentum method also extends to the case of nonholonomic systems. Building on the work on nonholonomic systems in Arnold [1988], Bates and Sniatycki [1993] and Bloch, Krishnaprasad, Mars- den and Murray [1996], on the example of the Routh problem in Zenkov [1995] and on the large Russian literature in this area, Zenkov, Bloch and Marsden [1998] show that there is a generalization to this setting. The method is effective in the sense that it applies to a wide variety of interesting examples, such as the rolling disk and a three wheeled vehicle known as the the roller racer. ........................... 15 July 1998—18h02 ...........................

56. 42 1. Introduction and Overview Exercises 1.7-1. Work out Cherry’s example of the Hamiltonian system in R4 whose energy function is given by (1.7.6). Show explicitly that the origin is a linearly and spectrally stable equilibrium but that it is nonlinearly unstable by proving that (1.7.7) is a solution for every τ > 0 which can be chosen to start arbitrarily close to the origin and which goes to inﬁnity for t → τ . 1.7-2. ˙ Show that (1.7.11) is Hamiltonian with p = M q, 1 1 H(q, p) = p · M −1 p + q · V q 2 2 and ∂F ∂K ∂K ∂F ∂F ∂K {F, K} = − i − S ij . ∂q i ∂pi ∂q ∂pi ∂pi ∂pj 1.7-3. Show that (up to an overall factor) the characteristic polynomial for the linear system (1.7.11) is p(λ) = det[λ2 M + λS + V ] and that this actually is a polynomial of degree n in λ2 . 1.7-4. Consider the two-degree of freedom system x − g y + γ x + αx = 0, ¨ ˙ ˙ y + g x + δ y + βy = 0. ¨ ˙ ˙ (a) Write it in the form (1.7.12). (b) For γ = δ = 0 show: (i) it is spectrally stable if α > 0, β > 0; (ii) for αβ < 0, it is spectrally unstable; (iii) for α < 0, β < 0, it is formally unstable (that is, the energy function, which is a quadratic form, is indeﬁnite); and A. if D := (g 2 + α + β)2 − 4αβ < 0, then there are two roots in the right half-plane and two in the left; the system is spectrally unstable; B. if D = 0 and g 2 + α + β ≥ 0 the system is spectrally stable, but if g 2 + α + β < 0 then it is spectrally unstable; and C. if D > 0 and g 2 + α + β ≥ 0 the system is spectrally stable, but if g 2 + α + β < 0 , then it is spectrally unstable. (c) For a polynomial p(λ) = λ4 + ρ1 λ3 + ρ2 λ2 + ρ3 λ + ρ4 , the Routh– Hurwitz criterion (see Gantmacher [1959], Volume 2)) says that the ........................... 15 July 1998—18h02 ...........................

57. 1.8 Bifurcation 43 number of right half-plane zeros of p is the number of sign changes of the sequence ρ1 ρ2 − ρ3 ρ3 ρ1 ρ2 − ρ2 − ρ4 ρ2 3 1 1, ρ1 , , , ρ4 . ρ1 ρ1 ρ2 − ρ3 Apply this to the case in which α < 0, β < 0, g 2 + α + β > 0, and at least one of γ or δ is positive to show that the system is spectrally unstable. 1.8 Bifurcation When the energy-momentum or energy-Casimir method indicates that an instability might be possible, techniques of bifurcation theory can be brought to bear to determine the emerging dynamical complexities such as the development of multiple equilibria and periodic orbits. Ball in a Rotating Hoop. For example, consider a particle moving with no friction in a rotating hoop (Figure 1.8.1). z ω R g = acceleration y due to gravity θ x Figure 1.8.1. A particle moving in a hoop rotating with angular velocity ω. In §2.8 we derive the equations and study the phase portraits for this system. One ﬁnds that as ω increases past g/R, the stable equilibrium at θ = 0 becomes unstable through a Hamiltonian pitchfork bifurcation and two new solutions are created. These solutions are symmetric in the vertical ........................... 15 July 1998—18h02 ...........................

58. 44 1. Introduction and Overview axis, a reflection of the original Z2 symmetry of the mechanical system in Figure 1.8.1. Breaking this symmetry by, for example, putting the rotation axis slightly off-center is an interesting topic that we shall discuss in §2.8. Rotating Liquid Drop. The system consists of the two-dimensional Euler equations for an ideal fluid with a free boundary. An equilibrium solution consists of a rigidly rotating circular drop. The energy-Casimir method shows stability provided that 3τ Ω<2 . (1.8.1) R3 In this formula, Ω is the angular velocity of the circular drop, R is its radius, and τ is the surface tension, a constant. As Ω increases and (1.8.1) is violated, the stability of the circular solution is lost and is picked up by elliptical-like solutions with Z2 × Z2 symmetry. The bifurcation is actually subcritical relative to Ω (that is, the new solutions occur below the critical value of Ω) and is supercritical (the new solutions occur above criticality) relative to the angular momentum. This is proved in Lewis, Marsden, and Ratiu [1987] and Lewis [1989], where other references may also be found (see Figure 1.8.2). increasing R angular momentum circular stable solutions uniformly rotating elliptical-like solutions Figure 1.8.2. A circular liquid drop losing its stability and its symmetry. For the ball in the hoop, the eigenvalue evolution for the linearized equations is shown in Figure 1.8.3(a). For the rotating liquid drop the movement of eigenvalues is the same: they are constrained to stay on the imaginary axis because of the symmetry of the problem. Without this symmetry, eigenvalues typically split, as in Figure 1.8.3(b). These are examples of a general theory of the movement of such eigenvalues given in Golubitsky and Stewart [1987], Dellnitz, Melbourne, and Marsden [1992], Knobloch, Mahalov, and Marsden [1994], and Kruse, Mahalov, and Marsden [1998]. More Examples. Another example is the heavy top: a rigid body with one point fixed, moving in a gravitational field. When the top makes the ........................... 15 July 1998—18h02 ...........................

59. 1.8 Bifurcation 45 y y C C x x (a) with symmetry (b) without symmetry Figure 1.8.3. The movement of eigenvalues in bifurcation of equilibria. transition from a fast top to a slow top, the angular velocity ω decreasess pas the critical value √ 2 M glI1 ωc = , (1.8.2) I3 stability is lost, and a resonance bifurcation occurs. Here, when the bifurcation occurs, the eigenvalues of the equations linearized at the equilibrium behave as in Figure 1.8.4. C y fast-slow y C transition x x Figure 1.8.4. Eigenvalue movement in the Hamiltonian Hopf bifurcation. For an extensive study of bifurcations and stability in the dynamics of a heavy top, see Lewis, Ratiu, Simo, and Marsden [1992]. Behavior of this sort is sometimes called a Hamiltonian Krein-Hopf bifurcation, or a ........................... 15 July 1998—18h02 ...........................

60. 46 1. Introduction and Overview gyroscopic instability (see Van der Meer [1985, 1990]). Here more complex dynamic behavior ensues, including periodic and chaotic motions (see Holmes and Marsden [1983]). In some systems with symmetry, the eigenvalues can pass as well as split, as has been shown by Dellnitz, Melbourne, and Marsden [1992] and references therein. More sophisticated examples, such as the dynamics of two coupled three- dimensional rigid bodies requires a systematic development of the basic theory of Golubitsky and Schaeﬀer [1985] and Golubitsky, Stewart, and Schaeﬀer [1988]. This theory is begun in, for example, Duistermaat [1983], Lewis, Marsden, and Ratiu [1987], Lewis [1989], Patrick [1989], Meyer and Hall [1992], Broer, Chow, Kim, and Vegter [1993], and Golubitsky, Mars- den, Stewart, and Dellnitz [1994]. For bifurcations in the double spherical pendulum (which includes a Hamiltonian-Krein-Hopf bifurcation), see Dell- nitz, Marsden, Melbourne, and Scheurle [1992] and Marsden and Scheurle [1993a]. Exercises 1.8-1. Study the bifurcations (changes in the phase portrait) for the equation x + µx + x2 = 0 ¨ as µ passes through zero. Use the second derivative test on the potential energy discussed in §1.10. 1.8-2. Repeat Exercise 1.8-1 for x + µx + x3 = 0 ¨ as µ passes through zero. 1.9 The Poincar´–Melnikov Method e The Forced Pendulum. To begin with a simple example, consider the equation of a forced pendulum ¨ φ + sin φ = cos ωt. (1.9.1) Here ω is a constant angular forcing frequency and is a small parameter. Systems of this or a similar nature arise in many interesting situations. For example, a double planar pendulum and other “executive toys” exhibit chaotic motion that is analogous to the behavior of this equation; see Burov [1986] and Shinbrot, Grebogi, Wisdom, and Yorke [1992]. For = 0 this has the phase portrait of a simple pendulum (the same as shown later in Figure 2.8.2a). For small but nonzero, (1.9.1) possesses no ........................... 15 July 1998—18h02 ...........................

61. 1.9 The Poincar´–Melnikov Method e 47 analytic integrals of the motion. In fact, it possesses transversal intersecting stable and unstable manifolds (separatrices); that is, the Poincar´ maps e Pt0 : R2 → R2 that advance solutions by one period T = 2π/ω starting at time t0 possess transversal homoclinic points. This type of dynamic behavior has several consequences, besides precluding the existence of analytic integrals, that lead one to use the term “chaotic.” For example, (1.9.1) has infinitely many periodic solutions of arbitrarily high period. Also, using the shadowing lemma, one sees that given any bi-infinite sequence of zeros and ones (for example, use the binary expansion of e or π), there exists a corresponding solution of (1.9.1) that successively crosses the plane φ = 0 (the pendulum’s vertically downward configuration) with φ > 0 corresponding to a zero and φ < 0 corresponding to a one. The origin of this chaos on an intuitive level lies in the motion of the pendulum near its unperturbed homoclinic orbit, the orbit that does one revolution in infinite time. Near the top of its motion (where φ = ±π) small nudges from the forcing term can cause the pendulum to fall to the left or right in a temporally complex way. The dynamical systems theory needed to justify the preceding statements is available in Smale [1967], Moser [1973], Guckenheimer and Holmes [1983], and Wiggins [1988, 1990]. Some key people responsible for the development of the basic theory are Poincar´, Birkhoff, Kolmogorov, Melnikov, Arnold, e Smale, and Moser. The idea of transversal intersecting separatrices comes from Poincar´’s famous paper on the three-body problem (Poincar´ [1890]). e e His goal, not quite achieved for reasons we shall comment on later, was to prove the nonintegrability of the restricted three body problem and that various series expansions used up to that point diverged (he began the theory of asymptotic expansions and dynamical systems in the course of this work). See Diacu and Homes [1996] for additional information about Poincar´’s work. e Although Poincar´ had all the essential tools needed to prove that equa- e tions like (1.9.1) are not integrable (in the sense of having no analytic integrals), his interests lay with harder problems and he did not develop the easier basic theory very much. Important contributions were made by Melnikov [1963] and Arnold [1964] which lead to a simple procedure for proving that (1.9.1) is not integrable. The Poincar´–Melnikov method was e revived by Chirikov [1979], Holmes [1980b] and Chow, Hale, and Mallet- Paret [1980]. We shall give the method for Hamiltonian systems. We refer to Guckenheimer and Holmes [1983] and to Wiggins [1988, 1990] for generalizations and further references. The Poincar´–Melnikov Method. e This method proceeds as follows: 1. Write the dynamical equation to be studied in the form x = X0 (x) + X1 (x, t), ˙ (1.9.2) ........................... 15 July 1998—18h02 ...........................

62. 48 1. Introduction and Overview where x ∈ R2 , X0 is a Hamiltonian vector field with energy H0 , X1 is periodic with period T and is Hamiltonian with energy a T -periodic function H1 . Assume that X0 has a homoclinic orbit x(t) so x(t) → x0 , a hyperbolic saddle point, as t → ±∞. 2. Compute the Poincar´–Melnikov function defined by e ∞ M (t0 ) = {H0 , H1 }(x(t − t0 ), t) dt (1.9.3) −∞ where { , } denotes the Poisson bracket. If M (t0 ) has simple zeros as a function of t0 , then (1.9.2) has, for sufficiently small , homoclinic chaos in the sense of transversal intersecting separatrices (in the sense of Poincar´ maps as mentioned e above). We shall prove this result in §2.11. To apply it to equation (1.9.1) one ˙ proceeds as follows. Let x = (φ, φ) so we get d φ ˙ φ 0 ˙ = + . dt φ − sin φ cos ωt The homoclinic orbits for = 0 are given by (see Exercise 1.9-1) φ(t) ±2 tan−1 (sinh t) x(t) = = ˙ φ(t) ±2 sech t and one has ˙ ˙ ˙ H0 (φ, φ) = 1 φ2 − cos φ and H1 (φ, φ, t) = φ cos ωt. (1.9.4) 2 Hence (1.9.3) gives ∞ ∂H0 ∂H1 ∂H0 ∂H1 M (t0 ) = − (x(t − t0 ), t) dt −∞ ˙ ∂φ ∂ φ ˙ ∂ φ ∂φ ∞ =− ˙ φ(t − t0 ) cos ωt dt −∞ ∞ = [2 sech(t − t0 ) cos ωt] dt. −∞ Changing variables and using the fact that sech is even and sin is odd, we get ∞ M (t0 ) = 2 sech t cos ωt dt cos(ωt0 ). −∞ ........................... 15 July 1998—18h02 ...........................

63. 1.10 Resonances, Geometric Phases, and Control 49 The integral is evaluated by residues (see Exercise 1.9-2): πω M (t0 ) = 2π sech cos(ωt0 ), (1.9.5) 2 which clearly has simple zeros. Thus, this equation has chaos for small enough. Exercises 1.9-1. Verify directly that the homoclinic orbits for the simple pendulum equation φ + sin φ = 0 are given by φ(t) = ±2 tan−1 (sinh t). ¨ ∞ 1.9-2. Evaluate the integral −∞ sech t cos ωt dt to prove (1.9.5) as follows. Write sech t = 2/(et + e−t ) and note that there is a simple pole of eiωz + e−iωz f (z) = ez + e−z in the complex plane at z = πi/2. Evaluate the residue there and apply Cauchy’s theorem 8 . 1.10 Resonances, Geometric Phases, and Control The work of Smale [1970] shows that topology plays an important role in mechanics. Smale’s work employs Morse theory applied to conserved quantities such as the energy-momentum map. In this section we point out other ways in which geometry and topology enter mechanical problems. The One-to-One Resonance. When one considers resonant systems one often encounters Hamiltonians of the form 1 2 λ 2 H= (q1 + p2 ) + (q2 + p2 ) + higher-order terms. 1 2 (1.10.1) 2 2 The quadratic terms describe two oscillators that have the same frequency when λ = 1, which is why one speaks of a one-to-one resonance. To analyze the dynamics of H, it is important to utilize a good geometric picture for the critical case 1 2 H0 = (q + p2 + q2 + p2 ). 2 (1.10.2) 2 1 1 2 8 Consult a book on complex variables such as Marsden and Hoﬀman, Basic Complex Analysis, Third Edition, Freeman, 1998. ........................... 15 July 1998—18h02 ...........................

64. 50 1. Introduction and Overview The energy level H0 = constant is the three-sphere S 3 ⊂ R4 . If we think of H0 as a function on C2 by letting z1 = q1 + ip1 and z2 = q2 + ip2 , then H0 = (|z1 |2 + |z2 |2 )/2 and so H0 is left-invariant by the action of SU(2), the group of complex 2 × 2 unitary matrices of determinant one. The corresponding conserved quantities are W1 = 2(q1 q2 + p1 p2 ), W2 = 2(q2 p1 − q1 p2 ), (1.10.3) W3 = 2 q1 + p2 1 − 2 q2 − p2 , 2 which comprise the components of a (momentum) map J : R 4 → R3 . (1.10.4) 2 2 2 2 From the relation 4H0 = W1 + W2 + W3 , one finds that J restricted to 3 S gives a map j : S3 → S2. (1.10.5) The fibers j −1 (point) are circles and the dynamics of H0 moves along these circles. The map j is the Hopf fibration which describes S 3 as a topologically nontrivial circle bundle over S 2 . The role of the Hopf fibration in mechanics was known to Reeb [1949]. One also finds that the study of systems like (1.10.1) that are close to H0 can, to a good approximation, be reduced to dynamics on S 2 . These dynamics are in fact Lie–Poisson and S 2 sits as a coadjoint orbit in so(3)∗ , so the evolution is of rigid body type, just with a different Hamiltonian. For a computer study of the Hopf fibration in the one-to-one resonance, see Kocak, Bisshopp, Banchoff, and Laidlaw [1986]. The Hopf Fibration in Rigid Body Mechanics. When doing reduction for the rigid body, one studies the reduced space J−1 (µ)/Gµ = J−1 (µ)/S 1 , which in this case is the sphere S 2 . Also, as we shall see in Chapter 15, J−1 (µ) is topologically the same as the rotation group SO(3), which in turn is the same as S 3 /Z2 . Thus, the reduction map is a map of SO(3) to S 2 . Such a map is given explicitly by taking an orthogonal matrix A and mapping it to the vector on the sphere given by Ak, where k is the unit vector along the z-axis. This map that does the projection is in fact a restriction of a momentum map and is, when composed with the map of S 3 ∼ SU(2) to SO(3), just the Hopf fibration again. Thus, not only = does the Hopf fibration occur in the one-to-one resonance, it occurs in the rigid body in a natural way as the reduction map from material to body representation! ........................... 15 July 1998—18h02 ...........................

65. 1.10 Resonances, Geometric Phases, and Control 51 Geometric Phases. The history of this concept is complex. We refer to Berry [1990] for a discussion of the history, going back to Bortolotti in 1926, Vladimirskii and Rytov in 1938 in the study of polarized light, to Kato in 1950 and Longuet-Higgins and others in 1958 in atomic physics. Some additional historical comments regarding phases in rigid body mechanics are given below. We pick up the story with the classical example of the Foucault pendulum. The Foucault pendulum gives an interesting phase shift (a shift in the angle of the plane of the pendulum’s swing) when the overall system undergoes a cyclic evolution (the pendulum is carried in a circular motion due to the Earth’s rotation). This phase shift is geometric in character: if one parallel transports an orthonormal frame along the same line of latitude, it returns with a phase shift equaling that of the Foucault pendulum. This phase shift ∆θ = 2π cos α (where α is the co-latitude) has the geometric meaning shown in Figure 1.10.1. cut and unroll cone parallel translate frame along a line of latitude Figure 1.10.1. The geometric interpretation of the Foucault pendulum phase shift. In geometry, when an orthonormal frame returns after traversing a closed path to its original position but rotated, the rotation is referred to as holonomy (or anholonomy ). This is a unifying mathematical concept that underlies many geometric phases in systems such as ﬁber optics, MRI (magnetic resonance imaging), amoeba propulsion, molecular dynamics, micromotors, and other eﬀects. These applications represent one reason why the subject is of such current interest. In the quantum case a seminal paper on geometric phases is Kato [1950]. It was Berry [1984, 1985], Simon [1984], Hannay [1985], and Berry and Hannay [1988] who realized that holonomy is the crucial geometric unify- ........................... 15 July 1998—18h02 ...........................

66. 52 1. Introduction and Overview ing thread. On the other hand, Golin, Knauf, and Marmi [1989], Mont- gomery [1988], and Marsden, Montgomery, and Ratiu [1989, 1990] demon- strated that averaging connections and reduction of mechanical systems with symmetry also plays an important role, both classically and quantum mechanically. Aharonov and Anandan [1987] have shown that the geometric phase for a closed loop in projectivized complex Hilbert space occurring in quantum mechanics equals the exponential of the symplectic area of a two-dimensional manifold whose boundary is the given loop. The symplectic form in question is naturally induced on the projective space from the canonical symplectic form of complex Hilbert space (minus the imaginary part of the inner product) via reduction. Marsden, Montgomery, and Ratiu [1990] show that this formula is the holonomy of the closed loop relative to a principal S 1 -connection on the unit ball of complex Hilbert space and is a particular case of the holonomy formula in principal bundles with abelian structure group. Geometric Phases and Locomotion. Geometric phases naturally occur is in families of integrable systems depending on parameters. Consider an integrable system with action-angle variables (I1 , I2 , . . . , In , θ1 , θ2 , . . . , θn ); assume the Hamiltonian H(I1 , I2 , . . . In ; m) depends on a parameter m ∈ M . This just means that we have a Hamiltonian independent of the angular variables θ and we can identify the configuration space with an n-torus Tn . Let c be a loop based at a point m0 in M . We want to compare the angular variables in the torus over m0 , once the system is slowly changed as the parameters undergo the circuit c. Since the dynamics in the fiber varies as we move along c, even if the actions vary by a negligible amount, there will be a shift in the angle variables due to the frequencies ω i = ∂H/∂I i of the integrable system; correspondingly, one defines 1 dynamic phase = ω i (I, c(t)) dt. 0 Here we assume that the loop is contained in a neighborhood whose standard action coordinates are defined. In completing the circuit c, we return to the same torus, so a comparison between the angles makes sense. The actual shift in the angular variables during the circuit is the dynamic phase plus a correction term called the geometric phase. One of the key results is that this geometric phase is the holonomy of an appropriately constructed connection called the Hannay-Berry connection on the torus bundle over M which is constructed from the action-angle variables. The corresponding angular shift, computed by Hannay [1985], is called Han- nay’s angles, so the actual phase shift is given by ∆θ = dynamic phases + Hannay’s angles. ........................... 15 July 1998—18h02 ...........................

67. 1.10 Resonances, Geometric Phases, and Control 53 The geometric construction of the Hannay-Berry connection for classical systems is given in terms of momentum maps and averaging in Golin, Knauf, and Marmi [1989] and Montgomery [1988]. Weinstein [1990] makes precise the geometric structures which make possible a definition of the Hannay angles for a cycle in the space of lagrangian submanifolds, even without the presence of an integrable system. Berry’s phase is then seen as a “primitive” for the Hannay angles. A summary of this work is given in Woodhouse [1992]. Another class of examples where geometric phases naturally arise is in the dynamics of coupled rigid bodies. The three dimensional single rigid body is discussed below. For several coupled rigid bodies, the dynamics can be quite complex. For instance, even for bodies in the plane, the dynamics is known to be chaotic, despite the presence of stable relative equilibria; see Oh, Sreenath, Krishnaprasad, and Marsden [1989]. Geometric phase phenomena for this type of example are quite interesting and are related to some of the work of Wilczek and Shapere on locomotion in micro-organisms. (See, for example, Shapere and Wilczek [1987, 1989] and Wilczek and Shapere [1989].) In this problem, control of the system’s internal variables can lead to phase changes in the external variables. These choices of variables are related to the variables in the reduced and the unreduced phase spaces. In this setting one can formulate interesting questions of optimal control such as “When a cat falls and turns itself over in mid-flight (all the time with zero angular momentum!) does it do so with optimal efficiency in terms of, say, energy expended?” There are interesting answers to these questions that are related to the dynamics of Yang–Mills particles moving in the associated gauge field of the problem. See Montgomery [1984, 1990] and references therein. We give two simple examples of how geometric phases for linked rigid bodies works. Additional details can be found in Marsden, Montgomery, and Ratiu [1990]. First, consider three uniform coupled bars (or coupled planar rigid bodies) linked together with pivot (or pin) joints, so the bars are free to rotate relative to each other. Assume the bars are moving freely in the plane with no external forces and that the angular momentum is zero. However, assume that the joint angles can be controlled with, say, motors in the joints. Figure 1.10.2 shows how the joints can be manipulated, each one going through an angle of 2π and yet the overall assemblage rotates through an angle π. Here we assume that the moments of inertia of the two outside bars (about an axis through their centers of mass and perpendicular to the page) are each one-half that of the middle bar. The statement is verified by examining the equation for zero angular momentum (see, for example Sreenath, Oh, Krishnaprasad, and Marsden [1988] and Oh, Sreenath, Krishnaprasad, and Marsden [1989]). General formulas for the reconstruction phase applicable to examples of this type are given in Krishnaprasad [1989]. ........................... 15 July 1998—18h02 ...........................

68. 54 1. Introduction and Overview A second example is the dynamics of linkages. This type of example is considered in Krishnaprasad [1989], Yang and Krishnaprasad [1990], including comments on the relation with the three-manifold theory of Thurston. Here one considers a linkage of rods, say four rods linked by pivot joints as in Figure 1.10.3. The system is free to rotate without external forces or torques, but there are assumed to be torques at the joints. When one turns the small “crank” the whole assemblage turns even though the angular momentum, as in the previous example, stays zero. Figure 1.10.2. Manipulating the joint angles can lead to an overall rotation of the system. For an overview of how geometric phases are used in robotic locomotion problems, see Marsden and Ostrowski [1998] (This paper is available at http://www.cds.caltech.edu/~marsden.) crank overall phase rotation of the assemblage Figure 1.10.3. Turning the crank can lead to an overall phase shift. Phases in Rigid Body Dynamics. As we shall see in Chapter 15, the motion of a rigid body is a geodesic with respect to a left-invariant ........................... 15 July 1998—18h02 ...........................

69. 1.10 Resonances, Geometric Phases, and Control 55 Riemannian metric (the inertia tensor) on SO(3). The corresponding phase space is P = T ∗ SO(3) and the momentum map J : P → R3 for the left SO(3) action is right translation to the identity. We identify so(3)∗ with so(3) via the Killing form and identify R3 with so(3) via the map v → v , ˆ where v (w) = v × w, × being the standard cross product. Points in so(3)∗ ˆ are regarded as the left reduction of T ∗ SO(3) by G = SO(3) and are the angular momenta as seen from a body-fixed frame. The reduced spaces Pµ = J−1 (µ)/Gµ are identified with spheres in R3 of Euclidean radius µ , with their symplectic form ωµ = −dS/ µ , where dS is the standard area form on a sphere of radius µ and where Gµ consists of rotations about the µ-axis. The trajectories of the reduced dynamics are obtained by intersecting a family of homothetic ellipsoids (the energy ellipsoids) with the angular momentum spheres. In particular, all but at most four of the reduced trajectories are periodic. These four exceptional trajectories are the well-known homoclinic trajectories; we shall determine them explicitly in §15.8. Suppose a reduced trajectory Π(t) is given on Pµ , with period T . Af- ter time T , by how much has the rigid body rotated in space? The spatial angular momentum is π = µ = gΠ, which is the conserved value of J. Here g ∈ SO(3) is the attitude of the rigid body and Π is the body angular momentum. If Π(0) = Π(T ), then µ = g(0)Π(0) = g(T )Π(T ) and so g(T )−1 µ = g(0)−1 µ, that is, g(T )g(0)−1 µ is a rotation about the axis µ. We want to give the angle of this rotation. To answer this question, let c(t) be the corresponding trajectory in J−1 (µ) ⊂ P . Identify T ∗ SO(3) with SO(3)×R3 by left trivialization, so c(t) gets identified with (g(t), Π(t)). Since the reduced trajectory Π(t) closes after time T , we recover the fact that c(T ) = gc(0) for some g ∈ Gµ . Here, g = g(T )g(0)−1 in the preceding notation. Thus, we can write g = exp[(∆θ)ζ], (1.10.6) where ζ = µ/ µ identifies gµ with R by aζ → a, for a ∈ R. Let D be one of the two spherical caps on S 2 enclosed by the reduced trajectory, let Λ be the corresponding oriented solid angle, that is, |Λ| = (area D)/ µ 2 , and let Hµ be the energy of the reduced trajectory. See Figure 1.10.4. All norms are taken relative to the Euclidean metric of R3 . Montgomery [1991a] and Marsden, Montgomery, and Ratiu [1990] show that modulo 2π, we have the rigid body phase formula: 1 2Hµ T ∆θ = ωµ + 2Hµ T = −Λ + . (1.10.7) µ D µ More History. The history of the rigid body phase formula is quite interesting and seems to have proceeded independently of the other de- ........................... 15 July 1998—18h02 ...........................

70. 56 1. Introduction and Overview true trajectory dynamic phase horizontal lift geometric phase πµ reduced trajectory D P Pµµ Figure 1.10.4. The geometry of the rigid body phase shift formula. velopments above9 . The formula has its roots in MacCullagh [1840] and Thomson and Tait [1867, §§123, 126]. (See Zhuravlev [1996] and O’Reilly [1997] for a discussion and extensions). A special case of formula (1.10.7) Update Refs is given in Ishlinskii [1952]; see also Ishlinskii [1963]. On page 195 of a later For Tudor: See Berceau’s book on mechanics, Ishlinskii [1976] notes that “the formula was found by papers the author in 1943 and was published in Ishlinskii [1952].” The formula referred to in the works of Ishlinskii covers a special case in which only the geometric phase is present. For example, in certain precessional motions in which, up to a certain order in averaging, one can ignore the dynamic phase and only the geometric phase survives. Even though Ishlinskii only found special cases of the result, he recognized that it is related to the geometric concept of parallel transport. A formula like the one above was found by Goodman and Robinson [1958] in the context of drift in gyroscopes; their proof is based on the Gauss-Bonnet theorem. Another interesting approach to formulas of this sort, also based on averaging and solid angles is given in Goldreich and Toomre [1969] who applied it to the interesting geophysical problem of polar wander (see also Poincar´ [1910]!). e 9 We thank V. Arnold for valuable help with these comments. ........................... 15 July 1998—18h02 ...........................

71. 1.10 Resonances, Geometric Phases, and Control 57 The special case of the above formula for a symmetric free rigid body was given by Hannay [1985] and Anandan [1988], formula (20). The proof of the general formula based on the theory of connections and the formula for holonomy in terms of curvature, was given by Montgomery [1991] and Marsden, Montgomery, and Ratiu [1990]. The approach using the Gauss- Bonnet theorem and its relation to the Poinsot construction along with additional results is taken up by Levi [1993]. For applications to general resonance problems (such as the three-wave interaction) and nonlinear optics, see Alber, Luther, Marsden, and Robbins [1998]. An analogue of the rigid body formula for the heavy top and the Lagrange top (symmetric heavy top) was given in Marsden, Montgomery, and Ratiu [1990]. Links with vortex filament configurations were given in Fukumoto and Miyajima [1996] and Fukumoto [1997]. Satellites with Rotors and Underwater Vehicles. Another example which naturally gives rise to geometric phases is the rigid body with one or more internal rotors. Figure 1.10.5 illustrates the system considered. rigid carrier spinning rotors Figure 1.10.5. The rigid body with internal rotors. To specify the position of this system we need an element of the group of rigid motions of R3 to place the center of mass and the attitude of the carrier, and an angle (element of S 1 ) to position each rotor. Thus the configuration space is Q = SE(3) × S 1 × S 1 × S 1 . The equations of motion of this system are an extension of Euler’s equations of motion for a free spinning rotor. Just as holding a spinning bicycle wheel while sitting on a swivel chair can affect the carrier’s motion, so the spinning rotors can affect the dynamics of the rigid carrier. In this example, one can analyze equilibria and their stability in much the same way as one can with the rigid body. However, what one often wants to ........................... 15 July 1998—18h02 ...........................

72. 58 1. Introduction and Overview do is to forcibly spin, or control, the rotors so that one can achieve attitude control of the structure in the same spirit that a falling cat has control of its attitude by manipulating its body parts while falling. For example, one can attempt to prescribe a relation between the rotor dynamics and the rigid body dynamics by means of a feedback law . This has the property that the total system angular momentum is still preserved and that the resulting dynamic equations can be expressed entirely in terms of the free rigid body variable. (A falling cat has zero angular momentum even though it is able to turn over!) In some cases the resulting equations are again Hamiltonian on the invariant momentum sphere. Using this fact, one can compute the geometric phase for the problem generalizing the free rigid body phase formula. (See Bloch, Krishnaprasad, Marsden, and Sńchez a [1992] and Bloch, Leonard, and Marsden [1997, 1998] for details.) One hopes that this type of analysis will be useful in designing and understanding attitude control devices. Another example that combines some features of the satellite and the heavy top is the underwater vehicle. This is in the realm of the dynamics of rigid bodies in fluids, a subject going back to Kirchoff in the late 1800’s. We refer to Leonard and Marsden [1997] and Holmes, Jenkins, and Leonard [1998] for modern accounts and many references. Miscellaneous Links. There are many continuum mechanical examples to which the techniques of geometric mechanics apply. Some of those are free boundary problems (Lewis, Marsden, Montgomery, and Ratiu [1986], Montgomery, Marsden, and Ratiu [1984], Mazer and Ratiu [1989]), spacecraft with flexible attachments (Krishnaprasad and Marsden [1987]), elasticity (Holm and Kupershmidt [1983], Kupershmidt and Ratiu [1983], Mars- den, Ratiu, and Weinstein [1984a,b], Simo, Marsden, and Krishnaprasad [1988]), and reduced MHD (Morrison and Hazeltine [1984] and Marsden and Morrison [1984]). We also wish to look at these theories from both the spatial (Eulerian) and body (convective) points of view as reductions of the canonical material picture. These two reductions are, in an appropriate sense, dual to each other. Reduction also finds use in a number of other diverse areas as well. We mention just a few samples. Update references • Integrable systems (Moser [1980], Perelomov [1990], Adams, Harnad, and Previato [1988], Fomenko and Trofimov [1989], Fomenko [1989], Reyman and Semenov–Tian–Shansky [1990] and Moser and Veselov [1990]). • Applications of integrable systems to numerical analysis (like the QR algorithm and sorting algorithms); see Deift and Li [1989] and Bloch, Brockett, and Ratiu [1990, 1992]. • Numerical integration, (Sanz-Serra and Calvo [1994], Marsden, Patrick, and Shadwick [1996], Wendlandt and Marsden [1977], Marsden, Patrick, ........................... 15 July 1998—18h02 ...........................

73. 1.10 Resonances, Geometric Phases, and Control 59 and Shkoller [1997]) • Hamiltonian chaos (Arnold [1964], Ziglin [1980a,b, 1981], Holmes and Marsden [1981, 1982a,b, 1983], Wiggins [1988]). • Averaging (Cushman and Rod [1982], Iwai [1982, 1985], Ercolani, Forest, McLaughlin, and Montgomery [1987]). • Hamiltonian bifurcations (Van der Meer [1985], Golubitsky and Scha- effer [1985], Golubitsky and Stewart [1987], Golubitsky, Stewart, and Schaeffer [1988], Lewis, Marsden, and Ratiu [1987], Lewis, Ratiu, Simo, and Marsden [1992], Montaldi, Roberts, and Stewart [1988], Golubitsky, Marsden, Stewart, and Dellnitz [1994]). • Algebraic geometry (Atiyah [1982, 1983], Kirwan [1984, 1985, 1988]). • Celestial mechanics (Deprit [1983], Meyer and Hall [1992]). • Vortex dynamics (Ziglin [1980b], Koiller, Soares, and Melo Neto [1985], Wan and Pulvirente [1984], Wan [1986, 1988a,b,c], Szeri and Holmes [1988]). • Solitons (Flaschka, Newell, and Ratiu [1983a,b], Newell [1985], Ko- vacic and Wiggins [1992], McLaughlin, Overman, Wiggins, and Xion [1993], Alber and Marsden [1992]). Xion? check index • Multisymplectic geometry, pde’s, and nonlinear waves (Gimmsy[1992], items---all 3 Bridges [1995,1996], Marsden and Shkoller [1997]). added • Relativity and Yang–Mills theory (Fischer and Marsden [1972, 1979], Arms [1981], Arms, Marsden, and Moncrief [1981, 1982]). • Fluid variational principles using Clebsch variables and “Lin constraints” (Seliger and Whitham [1968], Cendra and Marsden [1987], Cendra, Ibort, and Marsden [1987]). • Control, satellite and underwater vehicle dynamics (Krishnaprasad [1985], van der Shaft and Crouch [1987], Aeyels and Szafranski [1988], Bloch, Krishnaprasad, Marsden and Sńchez [1992], Wang, Krish- a naprasad and Maddocks [1991], Leonard [1997], Leonard and Mars- den [1997]), Bloch, Leonard, and Marsden [1998], and Holmes, Jenk- ins, and Leonard [1998]). • Nonholonomic systems (Naimark and Fufaev [1972], Koiller [1992], Bates and Sniatycki [1993], Bloch, Krishnaprasad, Marsden and Mur- ray [1996], Koon and Marsden [1997a,b,c]). ........................... 15 July 1998—18h02 ...........................

74. 60 1. Introduction and Overview Reduction is a natural historical culmination of the works of Liouville (for integrals in involution) and of Jacobi (for angular momentum) for reducing the phase space dimension in the presence of ﬁrst integrals. It is intimately connected with work on momentum maps and its forerunners appear already in Jacobi [1866], Lie [1890], Cartan [1922], and Whittaker [1927]. It was developed later in Kirillov [1962], Arnold [1966a], Kostant [1970], Souriau [1970], Smale [1970], Nekhoroshev [1977], Meyer [1973], and Marsden and Weinstein [1974]. See also Guillemin and Sternberg [1984] and Marsden and Ratiu [1986] for the Poisson case and Sjamaar and Lerman [1991] for the singular symplectic case. ........................... 15 July 1998—18h02 ...........................

75. This is page 61 Printer: Opaque this 2 Hamiltonian Systems on Linear Symplectic Spaces A natural arena for Hamiltonian mechanics is a symplectic or Poisson manifold. The first chapters concentrate on the symplectic case while Chapter 10 introduces the Poisson case. The symplectic context focuses on the symplectic two-form dq i ∧ dpi and its infinite-dimensional analogs, while the Poisson context looks at the Poisson bracket as the fundamental object. To facilitate the understanding of a number of points, we begin this chapter with the theory in linear spaces. This linear setting is already ad- equate for a number of interesting examples such as the wave equation and Schr¨dinger’s equation. Later in Chapter 4 we make the transition o to manifolds and in Chapters 7 and 8 we study the basics of Lagrangian mechanics. 2.1 Introduction To motivate the introduction of symplectic geometry in mechanics, we briefly recall from §1.1 the classical transition from Newton’s second law to the Lagrange and Hamilton equations. Newton’s Second Law for a particle moving in Euclidean three-space R3 , under the influence of a potential energy V (q), is F = ma, (2.1.1) where q ∈ R3 , F(q) = − V (q) is the force, m is the mass of the particle, and a = d2 q/dt2 is the acceleration (assuming we start in a postulated

76. 62 2. Hamiltonian Systems on Linear Symplectic Spaces privileged coordinate frame called an inertial frame).1 The potential energy V is introduced through the notion of work and the assumption that the force field is conservative. The introduction of the kinetic energy 2 1 dq K= m 2 dt is through the power , or rate of work equation: dK ˙ ¨ = m q, q = q, F ,˙ dt where , denotes the inner product on R3 . The Lagrangian is defined by m L(q i , q i ) = ˙ q 2 − V (q) ˙ (2.1.2) 2 and one checks by direct calculation that Newton’s second law is equivalent to the Euler–Lagrange equations: d ∂L ∂L − i = 0, (2.1.3) dt ∂ q i ˙ ∂q which are second-order differential equations in q i ; the equations (2.1.3) are worthy of independent study for a general L since they are the equations for stationary values of the action integral : t2 δ L(q i , q i ) dt = 0 ˙ (2.1.4) t1 as will be detailed later. These variational principles play a fundamental role throughout mechanics—both in particle mechanics and field theory. It is easily verified that dE/dt = 0, where E is the total energy: 1 m q 2 + V (q). E=˙ 2 Lagrange and Hamilton observed that it is convenient to introduce the momentum pi = mq i and rewrite E as a function of pi and q i by letting ˙ p 2 H(q, p) =+ V (q), (2.1.5) 2m for then Newton’s second law is equivalent to Hamilton’s canonical equations ∂H ∂H qi = ˙ , pi = − ˙ , (2.1.6) ∂pi ∂q i which is a first-order system in (q, p)-space, or phase space. 1 Newton and subsequent workers in mechanics thought of this inertial frame as one “fixed relative to the distant stars.” While this raises serious questions about what this could really mean mathematically or physically, it remains a good starting point. Deeper insight is found in Chapter 8 and in courses in general relativity. ........................... 15 July 1998—18h02 ...........................

77. 2.1 Introduction 63 Matrix Notation. For a deeper understanding of Hamilton’s equations, we recall some matrix notation (see Abraham, Marsden, and Ratiu [1988], §5.1 for more details). Let E be a real vector space and E ∗ its dual space. Let e1 , . . . , en be a basis of E with the associated dual basis for E ∗ denoted e1 , . . . , en ; that is, ei is defined by ei , ej := ei (ej ) = δj , i which equals 1 if i = j and 0 if i = j. Vectors v ∈ E are written v = v i ei (a sum on i is understood) and covectors α ∈ E ∗ as α = αi ei ; v i and αi are the components of v and α respectively. If A : E → F is a linear transformation, its matrix relative to bases e1 , . . . , en of E and f1 , . . . , fm of F is denoted Aji and is defined by A(ei ) = Aji fj , i.e., [A(v)]j = Aji v i . (2.1.7) Thus, the columns of the matrix of A are A(e1 ), . . . , A(en ); the upper index is the row index and the lower index is the column index. For other linear transformations, we place the indices in their corresponding places. For example, if A : E ∗ → F is a linear transformation, its matrix Aij satisfies A(ej ) = Aij fi , that is, [A(α)]i = Aij αj . If B : E × F → R is a bilinear form, its matrix Bij is defined by Bij = B(ei , fj ); i.e., B(v, w) = v i Bij wj . (2.1.8) Define the associated linear map B : E → F ∗ by B (v)(w) = B(v, w) and observe that B (ei ) = Bij f j . Since B (ei ) is the ith column of the matrix representing the linear map B , it follows that the matrix of B in the bases e1 , . . . , en , f 1 , . . . , f n is the transpose of Bij that is, [B ]ji = Bij . (2.1.9) Let Z denote the vector space of (q, p)’s and write z = (q, p). Let the coordinates q j , pj be collectively denoted by z I , I = 1, . . . , 2n. One reason for the notation z is that if one thinks of z as a complex variable z = q + ip, then Hamilton’s equations are equivalent to the following complex form of Hamilton’s equations (see Exercise 2.1-1): ∂H z = −2i ˙ , (2.1.10) ∂z where ∂/∂z := (∂/∂q − i∂/∂p)/2. ........................... 15 July 1998—18h02 ...........................

78. 64 2. Hamiltonian Systems on Linear Symplectic Spaces Symplectic and Poisson Structures. We can view Hamilton’s equations (2.1.6) as follows. Think of the operation ∂H ∂H ∂H ∂H dH(z) = , → ,− i =: XH (z), (2.1.11) ∂q i ∂pi ∂pi ∂q which forms a vector field XH , called the Hamiltonian vector field , from the differential of H, as the composition of the linear map R : Z∗ → Z with the differntial dH(z) of H. The matrix of R is 0 l [RAB ] = =: J, (2.1.12) −l 0 where we write J for the specific matrix (2.1.12) sometimes called the symplectic matrix . Thus, XH (z) = R · dH(z) (2.1.13) or, if the components of XH are denoted X I , I = 1, . . . , 2n, ∂H X I = RIJ , i.e., XH = J H (2.1.14) ∂z J where H is the naive gradient of H; that is, the row vector dH but regarded as a column vector. Let B(α, β) = α, R(β) be the bilinear form associated to R, where , denotes the canonical pairing between Z ∗ and Z. One calls either the bilinear form B or its associated linear map R, the Poisson structure. The classical Poisson bracket (consistent with what we defined in Chapter 1) is defined by {F, G} = B(dF, dG) = dF · J G. (2.1.15) The symplectic structure Ω is the bilinear form associated to R−1 : Z → Z ∗ , that is, Ω(v, w) = R−1 (v), w or, equivalently, Ω = R−1 . The matrix of Ω is J in the sense that Ω(v, w) = v T Jw. (2.1.16) To unify notation we shall sometimes write Ω for the symplectic form, Z ×Z →R with matrix J, Ω for the associated linear map, Z → Z∗ with matrix JT , Ω for the inverse map (Ω )−1 = R, Z∗ → Z with matrix J, B for the Poisson form, Z∗ × Z∗ → R with matrix J. ........................... 15 July 1998—18h02 ...........................

79. 2.2 Symplectic Forms on Vector Spaces 65 Hamilton’s equations may be written z = XH (z) = Ω dH(z). ˙ (2.1.17) Multiplying both sides by Ω , we get Ω XH (z) = dH(z). (2.1.18) In terms of the symplectic form, (2.1.18) reads Ω(XH (z), v) = dH(z) · v (2.1.19) for all z, v ∈ Z. Problems such as rigid body dynamics, quantum mechanics as a Hamil- tonian system, and the motion of a particle in a rotating reference frame motivate the need to generalize these concepts. We shall do this in subsequent chapters and deal with both symplectic and Poisson structures in due course. Exercises 2.1-1. Write z = q+ip and show that Hamilton’s equations are equivalent to ∂H z = −2i ˙ . ∂z Give a plausible definition of the right-hand side as part of your answer and recognize the usual formula from complex variable theory. 2.1-2. Write the harmonic oscillator m¨ + kx = 0 in the form of Euler– x Lagrange equations, as Hamilton’s equations, and finally, in the complex form (2.1.10). 2.1-3. Repeat Exercise 2.1-2 for m¨ + kx + αx3 = 0. x 2.2 Symplectic Forms on Vector Spaces Let Z be a real Banach space, possibly infinite dimensional, and let Ω : Z × Z → R be a continuous bilinear form on Z. The form Ω is said to be nondegenerate (or weakly nondegenerate) if Ω(z1 , z2 ) = 0 for all z2 ∈ Z implies z1 = 0. As in §2.1, the induced continuous linear mapping Ω : Z → Z ∗ is defined by Ω (z1 )(z2 ) = Ω(z1 , z2 ). (2.2.1) Nondegeneracy of Ω is equivalent to injectivity of Ω ; that is, to the condition “Ω (z) = 0 implies z = 0.” The form Ω is said to be strongly ........................... 15 July 1998—18h02 ...........................

80. 66 2. Hamiltonian Systems on Linear Symplectic Spaces nondegenerate if Ω is an isomorphism, that is, Ω is onto as well as being injective. The open mapping theorem guarantees that if Z is a Banach space and Ω is one-to-one and onto, then its inverse is continuous. In most of the infinite-dimensional examples discussed in this book Ω will be only (weakly) nondegenerate. A linear map between finite-dimensional spaces of the same dimension is one-to-one if and only if it is onto. Hence, when Z is finite dimensional, weak nondegeneracy and strong nondegeneracy are equivalent. If Z is finite dimensional, the matrix elements of Ω relative to a basis {eI } are defined by ΩIJ = Ω(eI , eJ ). If {eJ } denotes the basis for Z ∗ that is dual to {eI }, that is, eJ , eI = δI J I I and if we write z = z eI and w = w eI , then Ω(z, w) = z I ΩIJ wJ (sum over I, J). Since the matrix of Ω relative to the bases {eI } and {eJ } equals the transpose of the matrix of Ω relative to {eI }; that is (Ω )JI = ΩIJ , nondegeneracy is equivalent to det[ΩIJ ]= 0. In particular, if Ω is skew and nondegenerate, then Z is even dimensional, since the determinant of a skew-symmetric matrix with an odd number of rows (and columns) is zero. Definition 2.2.1. A symplectic form Ω on a vector space Z is a nondegenerate skew-symmetric bilinear form on Z. The pair (Z, Ω) is called a symplectic vector space. If Ω is strongly nondegenerate, (Z, Ω) is called a strong symplectic vector space. Examples We now develop some basic examples of symplectic forms. (a) Canonical Forms. Let W be a vector space, and let Z = W × W ∗ . Define the canonical symplectic form Ω on Z by Ω((w1 , α1 ), (w2 , α2 )) = α2 (w1 ) − α1 (w2 ), (2.2.2) where w1 , w2 ∈ W and α1 , α2 ∈ W ∗ . More generally, let W and W be two vector spaces in duality, that is, there is a weakly nondegenerate pairing , : W × W → R. Then on W ×W , Ω((w1 , α1 ), (w2 , α2 )) = α2 , w1 − α1 , w2 (2.2.3) is a weak symplectic form. ........................... 15 July 1998—18h02 ...........................

81. 2.2 Symplectic Forms on Vector Spaces 67 (b) The Space of Functions. Let F(R3 ) be the space of smooth functions ϕ : R3 → R, and let Denc (R3 ) be the space of smooth densities on R3 with compact support. We write a density π ∈ Denc (R3 ) as a function π ∈ F(R3 ) with compact support times the volume element d3 x on R3 as π = π d3 x. The spaces F and Denc are in weak nondegenerate duality by the pairing ϕ, π = ϕπ d3 x. Therefore, from (2.2.3), we get the symplectic form Ω on the vector space Z = F(R3 ) × Denc (R3 ): Ω((ϕ1 , π1 ), (ϕ2 , π2 )) = ϕ 1 π2 − ϕ 2 π1 . (2.2.4) R3 R3 We choose densities with compact support so that the integrals in this formula will be finite. Other choices of spaces could be used as well. (c) Finite-Dimensional Canonical Form. Suppose that W is a real vector space of dimension n. Let {ei } be a basis of W , and let {ei } be the dual basis of W ∗ . With Z = W × W ∗ and defining Ω : Z × Z → R as in (2.2.2), one computes that the matrix of Ω in the basis {(e1 , 0), . . . , (en , 0), (0, e1 ), . . . , (0, en )} is 0 l J= , (2.2.5) −l 0 where l and 0 are the n × n identity and zero matrices. (d) Symplectic Form Associated to an Inner Product Space. If (W, , ) is a real inner product space, W is in duality with itself, so we obtain a symplectic form on Z = W × W from (2.2.3): Ω((w1 , w2 ), (z1 , z2 )) = z2 , w1 − z1 , w2 . (2.2.6) As a special case of (2.2.6), let W = R3 with the usual inner product 3 q, v = q · v = qi vi . i=1 The corresponding symplectic form on R is given by 6 Ω((q1 , v1 ), (q2 , v2 )) = v2 · q1 − v1 · q2 , (2.2.7) where q1 , q2 , v1 , v2 ∈ R3 . This coincides with Ω defined in Example (c) for W = R3 , provided R3 is identified with (R3 )∗ . Bringing Ω to canonical form using elementary linear algebra results in the following statement. If (Z, Ω) is a p-dimensional symplectic vector space, then p is even. Furthermore, Z is isomorphic to W ×W ∗ and there is a basis of W in which the matrix of Ω is J. Such a basis is called canonical , as are the corresponding coordinates. See Exercise 2.2-3. ........................... 15 July 1998—18h02 ...........................

82. 68 2. Hamiltonian Systems on Linear Symplectic Spaces (e) Symplectic Form on Cn . Write elements of complex n-space Cn as n-tuples z = (z1 , . . . , zn ) of complex numbers. The Hermitian inner product is n n n z, w = zj wj = (xj uj + yj vj ) + i (uj yj − vj xj ), j=1 j=1 j=1 where zj = xj + iyj and wj = uj + ivj . Thus, Re z, w is the real inner product and − Im z, w is the symplectic form if Cn is identified with R n × Rn . (f ) Quantum Mechanical Symplectic Form. The following symplectic vector space arises in quantum mechanics, as we shall explain in Chap- ter 3. Recall that a Hermitian inner product , : H × H → C on a complex Hilbert space H is linear in its first argument, antilinear in its second, and ψ1 , ψ2 is the complex conjugate of ψ2 , ψ1 , where ψ1 , ψ2 ∈ H. Set Ω(ψ1 , ψ2 ) = −2 Im ψ1 , ψ2 , where is Planck’s constant. One checks that Ω is a strong symplectic form on H. Let H be the complexification of a real Hilbert space H, so it is identified with H × H, and the inner product is given by (u1 , u2 ), (v1 , v2 ) = u1 , v1 + u2 , v2 + i( u2 , v1 − u1 , v2 ). This form coincides with 2 times that in (2.2.6). On the other hand, if we embed H into H × H∗ via ψ → (iψ, ψ) then the restriction of times the canonical symplectic form (2.2.6) on H × H∗ , namely, ((ψ1 , ϕ1 ), (ψ2 , ϕ2 )) → Re[ ϕ2 , ψ1 − ϕ1 , ψ2 ], coincides with Ω . Exercises 2.2-1. Verify that the formula for the symplectic form for R2n as a matrix, namely, 0 l J= −l 0 coincides with the definition of the symplectic form as the canonical form on R2n regarded as the product Rn × (Rn )∗ . 2.2-2. Let (Z, Ω) be a finite-dimensional symplectic vector space and let V ⊂ Z be a linear subspace. Assume that V is symplectic; that is, Ω restricted to V × V is nondegenerate. Let V Ω = {z ∈ Z | Ω(z, v) = 0 for all v ∈ V }. Show that V Ω is symplectic and Z = V ⊕ V Ω . ........................... 15 July 1998—18h02 ...........................

83. 2.3 Canonical Transformations or Symplectic Maps 69 2.2-3. Find a canonical basis for a symplectic form Ω on Z as follows. Let e1 ∈ Z, e1 = 0. Find e2 ∈ Z with Ω(e1 , e2 ) = 0. By rescaling e2 , assume Ω(e1 , e2 ) = 1. Let V be the span of e1 and e2 . Apply Exercise 2.2-2 and repeat this construction on V Ω . 2.2-4. Let (Z, Ω) be a finite dimensional symplectic vector space and V ⊂ Z a subspace. Define V Ω as in Exercise 2.2-2. Show that Z/V Ω and V ∗ are isomorphic vector spaces. 2.3 Canonical Transformations or Symplectic Maps To motivate the definition of symplectic maps (synonymous with canonical transformations), start with Hamilton’s equations: ∂H ∂H qi = ˙ , pi = − ˙ , (2.3.1) ∂pi ∂q i and a transformation ϕ : Z → Z of phase space to itself. Write (˜, p) = ϕ(q, p) q ˜ that is, z = ϕ(z). ˜ (2.3.2) Assume z(t) = (q(t), p(t)) satisfies Hamilton’s equations, that is, z(t) = XH (z(t)) = Ω dH(z(t)), ˙ (2.3.3) where Ω : Z ∗ → Z is the linear map with matrix J whose entries we denote B JK . By the chain rule, z = ϕ(z) satisfies ˜ ˙I ∂ϕI J z = ˜ z =: AIJ z J ˙ ˙ (2.3.4) ∂z J (sum on J). Substituting (2.3.3) into (2.3.4), employing coordinate notation, and using the chain rule, we conclude that ˙I ∂H ∂H z = AIJ B JK K = AIJ B JK AL ˜ K . (2.3.5) ∂z ∂zL ˜ Thus, the equations (2.3.5) are Hamiltonian if and only if AIJ B JK AL = B IL , K (2.3.6) ........................... 15 July 1998—18h02 ...........................

84. 70 2. Hamiltonian Systems on Linear Symplectic Spaces or in matrix notation AJAT = J. (2.3.7) In terms of composition of linear maps, (2.3.6) means A ◦ Ω ◦ AT = Ω , (2.3.8) since the matrix of Ω in canonical coordinates is J (see §2.1). A transformation satisfying (2.3.6) is called a canonical transformation, a symplectic transformation, or a Poisson transformation 2 . Taking determinants of (2.3.7), shows that det A = ±1 (we will see in Chapter 9 that det A = 1 is the only possibility) and in particular that A is invertible; taking the inverse of (2.3.8) gives (AT )−1 ◦ Ω ◦ A−1 = Ω , that is, AT ◦ Ω ◦ A = Ω , (2.3.9) which has the matrix form AT JA = J (2.3.10) since the matrix of Ω in canonical coordinates is −J (see §2.1). Note that (2.3.7) and (2.3.10) are equivalent (the inverse of one gives the other). As bilinear forms, (2.3.9) reads Ω(Dϕ(z) · z1 , Dϕ(z) · z2 ) = Ω(z1 , z2 ), (2.3.11) where Dϕ is the derivative of ϕ (the Jacobian matrix in finite dimensions). With (2.3.11) as a guideline, we write the general condition for map to be symplectic. Definition 2.3.1. If (Z, Ω) and (Y, Ξ) are symplectic vector spaces, a smooth map f : Z → Y is called symplectic or canonical if it preserves the symplectic forms, that is, if Ξ(Df (z) · z1 , Df (z) · z2 ) = Ω(z1 , z2 ) (2.3.12) for all z, z1 , z2 ∈ Z. There is some notation that will help us write (2.3.12) in a compact and efficient way. 2 In Chapter 10, where Poisson structures can be different from symplectic ones, we will see that (2.3.8) generalizes to the Poisson context. ........................... 15 July 1998—18h02 ...........................

85. 2.3 Canonical Transformations or Symplectic Maps 71 Pull Back Notation We introduce a convenient notation for these sorts of transformations. ϕ∗ f pull back of a function: ϕ∗ f = f ◦ ϕ. ϕ∗ g push forward of a function: ϕ∗ g = g ◦ ϕ−1 . ϕ∗ X push forward of a vector field X by ϕ: (ϕ∗ X)(ϕ(z)) = Dϕ(z) · X(z); in components, ∂ϕI J (ϕ∗ X)I = X . ∂z J ϕ∗ Y pull back of a vector field Y by ϕ: ϕ∗ Y = (ϕ−1 )∗ Y ϕ∗ Ω pull back of a bilinear form Ω on Z gives a bilinear form ϕ∗ Ω depending on the point z ∈ Z: (ϕ∗ Ω)z (z1 , z2 ) = Ω(Dϕ(z) · z1 , Dϕ(z) · z2 ); in components, ∂ϕK ∂ϕL (ϕ∗ Ω)IJ = ΩKL ; ∂z I ∂z J ϕ∗ Ξ push forward a bilinear form Ξ by ϕ equals pull back by the inverse: ϕ∗ Ξ = (ϕ−1 )∗ Ξ. In this pull-back notation, (2.3.12) reads (f ∗ Ξ)z = Ωz , or f ∗ Ξ = Ω for short. The Symplectic Group. It is simple to verify that if (Z, Ω) is a finite- dimensional symplectic vector space, the set of all linear symplectic mappings T : Z → Z forms a group under composition. It is called the symplectic group and is denoted by Sp(Z, Ω). As we have seen, in a canonical basis, a matrix A is symplectic if and only if AT JA = J, (2.3.13) where AT is the transpose of A. For Z = W × W ∗ and a canonical basis, if A has the matrix Aqq Aqp A= , (2.3.14) Apq App then one checks (Exercise 2.3-2) that (2.3.13) is equivalent to either of the two conditions: ........................... 15 July 1998—18h02 ...........................

86. 72 2. Hamiltonian Systems on Linear Symplectic Spaces (1) Aqq AT and App AT are symmetric and Aqq AT − Aqp AT = l, qp pq pp pq (2) AT Aqq and AT App are symmetric and AT App − AT Apq = l. pq qp qq pq In infinite dimensions Sp(Z, Ω) is, by definition, the set of elements of GL(Z) (the group of invertible bounded linear operators of Z to Z ) that leave Ω fixed. Symplectic Orthogonal Complements. If (Z, Ω) is a (weak) symplectic space and E and F are subspaces of Z, we define E Ω = {z ∈ Z | Ω(z, e) = 0 for all e ∈ E}, called the symplectic orthogonal complement of E. We leave it to the reader to check that (i) E Ω is closed; (ii) E ⊂ F implies F Ω ⊂ E Ω ; (iii) E Ω ∩ F Ω = (E + F )Ω ; (iv) if Z is finite dimensional, then dim E +dim E Ω = dim Z (to show this, use the fact that E Ω = ker(i∗ ◦ Ω ), where i : E → Z is the inclusion and i∗ : Z ∗ → E ∗ is its dual, i∗ (α) = α ◦ i, which is surjective; alternatively, use Exercise 2.2-4); (v) if Z is finite dimensional, E ΩΩ = E (this is also true in infinite dimensions if E is closed); and (vi) if E and F are closed, then (E ∩ F )Ω = E Ω + F Ω (to prove this use iii and v). Exercises 2.3-1. Show that a transformation ϕ : R2n → R2n is symplectic in the sense that its derivative matrix A = Dϕ(z) satisfies the condition AT JA = J if and only if the condition Ω(Az1 , Az2 ) = Ω(z1 , z2 ) holds for all z1 , z2 ∈ R2n . 2.3-2. Let Z = W × W ∗ , let A : Z → Z be a linear transformation and, using canonical coordinates, write the matrix of A as Aqq Aqp A= . Apq App Show that A being symplectic is equivalent to either of the two conditions: (i) Aqq AT and App AT are symmetric and Aqq AT − Aqp AT = l; or qp pq pp pq ........................... 15 July 1998—18h02 ...........................

87. 2.4 The General Hamilton Equations 73 (ii) AT Aqq and AT App are symmetric and AT App − AT Aqp = l. (Here, pq qp qq pq l is the n × n identity.) 2.3-3. Let f be a given function of q = (q 1 , q 2 , . . . , q n ). Define the map ϕ : R2n → R2n by ϕ(q, p) = (q, p + df (q)). Show that ϕ is a canonical (symplectic) transformation. 2.3-4. (a) Let A ∈ GL(n, R) be an invertible linear transformation. Show that the map ϕ : R2n → R2n given by (q, p) → (Aq, (A−1 )T p) is a canonical transformation. (b) If R is a rotation in R3 , show that the map (q, p) → (Rq, Rp) is a canonical transformation. 2.3-5. Let (Z, Ω) be a finite dimensional symplectic vector space. A subspace E ⊂ Z is called isotropic, coisotroipic, and Lagrangian if E ⊂ E Ω , E Ω ⊂ E, and E = E Ω respectively. Note that, E is Lagrangian if and only if it is isotropic and coisotropic at the same time. Show that: (a) An isotropic (coisotropic) subspace E is Lagrangian if and only if dim E = dim E Ω . In this case necessarily 2 dim E = dim Z. (b) An isotropic (coisotropic) subspace is Lagrangian if and only if it is a maximal isotropic (minimal coisotropic) subspace. (c) Every isotropic (coisotropic) subspace is contained in (contains) a Lagrangian subspace. 2.4 The General Hamilton Equations The concrete form of Hamilton’s equations we have already encountered is a special case of a construction on symplectic spaces. Here, we discuss this formulation for systems whose phase space is linear; in subsequent sections we will generalize the setting to phase spaces which are symplectic manifolds and in Chapter 10 to spaces where only a Poisson bracket is given. These generalizations will all be important in our study of specific examples. Definition 2.4.1. Let (Z, Ω) be a symplectic vector space. A vector field X : Z → Z is called Hamiltonian if Ω (X(z)) = dH(z), (2.4.1) for all z ∈ Z, for some C 1 function H : Z → R. Here dH(z) = DH(z) is alternative notation for the derivative of H. If such an H exists, we write X = XH and call H a Hamiltonian function, or energy function for the vector field X. ........................... 15 July 1998—18h02 ...........................

88. 74 2. Hamiltonian Systems on Linear Symplectic Spaces In a number of important examples, especially infinite-dimensional ones, H need not be defined on all of Z. We shall briefly discuss some of the technicalities involved in §3.3. If Z is finite dimensional, nondegeneracy of Ω implies that Ω : Z → Z ∗ is an isomorphism, which guarantees that XH exists for any given function H. However, if Z is infinite dimensional and Ω is only weakly nondegenerate, we do not know a priori that XH exists for a given H. If it does exist, it is unique since Ω is one-to-one. The set of Hamiltonian vector fields on Z is denoted XHam (Z), or simply XHam . Thus XH ∈ XHam is the vector field determined by the condition Ω(XH (z), δz) = dH(z) · δz for all z, δz ∈ Z. (2.4.2) If X is a vector field, the interior product iX Ω is defined to be the dual vector (also called, a one form) given at a point z ∈ Z as follows: (iX Ω)z ∈ Z ∗ ; (iX Ω)z (v) := Ω(X(z), v), for all v ∈ Z. Then condition (2.4.1) or (2.4.2) may be written as iX Ω = dH; i.e., X Ω = dH. (2.4.3) To express H in terms of XH and Ω, we integrate the identity dH(tz) · z = Ω(XH (tz), z) from t = 0 to t = 1. The fundamental theorem of calculus gives 1 1 dH(tz) H(z) − H(0) = dt = dH(tz) · z dt 0 dt 0 1 = Ω(XH (tz), z) dt. (2.4.4) 0 Let us now abstract the calculation we did in arriving at (2.3.7). Proposition 2.4.2. Let (Z, Ω) and (Y, Ξ) be symplectic vector spaces and f : Z → Y a diffeomorphism. Then f is a symplectic transformation if and only if for all Hamiltonian vector fields XH on Y , we have f∗ XH◦f = XH ; that is, Df (z) · XH◦f (z) = XH (f (z)). (2.4.5) Proof. Note that for v ∈ Z, Ω(XH◦f (z), v) = d(H ◦ f )(z) · v = dH(f (z)) · Df (z) · v = Ξ(XH (f (z)), Df (z) · v). (2.4.6) ........................... 15 July 1998—18h02 ...........................

89. 2.4 The General Hamilton Equations 75 If f is symplectic, then Ξ(Df (z) · XH◦f (z), Df (z) · v) = Ω(XH◦f (z), v) and thus by nondegeneracy of Ξ and the fact that Df (z) · v is an arbitrary element of Y (because f is a diffeomorphism and hence Df (z) is an ismorphism), (2.4.5) holds. Conversely, if (2.4.5) holds, then (2.4.6) implies that Ξ(Df (z) · XH◦f (z), Df (z) · v) = Ω(XH◦f (z), v) for any v ∈ Z and any C 1 map H : Y → R. However, XH◦f (z) equals an arbitrary element w ∈ Z for a correct choice of the Hamiltonian function H, namely, (H ◦ f )(z) = Ω(w, z). Thus, f is symplectic. Definition 2.4.3. Hamilton’s equations for H is the system of differential equations defined by XH . Letting c : R → Z be a curve, they are the equations dc(t) = XH (c(t)). (2.4.7) dt The Classical Hamilton Equations. We now relate the abstract form (2.4.7) to the classical form of Hamilton’s equations. In the following, an n-tuple (q 1 , . . . , q n ) will be denoted simply by (q i ), etc. Proposition 2.4.4. Suppose that (Z, Ω) is a 2n-dimensional symplectic vector space, and let (q i , pi ) = (q 1 , . . . , q n , p1 , . . . , pn ) denote canonical coordinates, with respect to which Ω has matrix J. Then in this coordinate system, XH : Z → Z is given by ∂H ∂H XH = ,− i =J· H. (2.4.8) ∂pi ∂q Thus, Hamilton’s equations in canonical coordinates are dq i ∂H dpi ∂H = , =− i. (2.4.9) dt ∂pi dt ∂q More generally, if Z = V ×V , · , · : V ×V → R is a weakly nondegenerate pairing, and Ω((e1 , α1 ), (e2 , α2 )) = α2 , e1 − α1 , e2 , then δH δH XH (e, α) = ,− , (2.4.10) δα δe where δH/δα ∈ V and δH/δe ∈ V are the partial functional derivatives defined by δH D2 H(e, α) · β = β, (2.4.11) δα for any β ∈ V and similarly for δH/δe; in (2.4.10) it is assumed that the functional derivatives exist. ........................... 15 July 1998—18h02 ...........................

90. 76 2. Hamiltonian Systems on Linear Symplectic Spaces Proof. If (f, β) ∈ V × V , then δH δH δH δH Ω ,− , (f, β) = β, + ,f δα δe δα δe = D2 H(e, α) · β + D1 H(e, α) · f = dH(e, α), (f, β) . Proposition 2.4.5. (Conservation of Energy) Let c(t) be an integral curve of XH . Then H(c(t)) is constant in t. If ϕt denotes the flow of XH , that is, ϕt (z) is the solution of (2.4.7) with initial conditions z ∈ Z, then H ◦ ϕt = H. Proof. By the chain rule, d d d H(c(t)) = dH(c(t)) · c(t) = Ω XH (c(t)), c(t) dt dt dt = Ω (XH (c(t)), XH (c(t))) = 0, where the final equality follows from the skew-symmetry of Ω. Exercises 2.4-1. Let the skew-symmetric bilinear form Ω on R2n have the matrix B l , −l 0 where B = [Bij ] is a skew-symmetric n × n matrix, and 1 is the identity matrix. (a) Show that Ω is nondegenerate and hence a symplectic form on R2n . (b) Show that Hamilton’s equations with respect to Ω are, in standard coordinates, dq i ∂H dpi ∂H ∂H = , = − i − Bij . dt ∂pi dt ∂q ∂pj 2.5 When Are Equations Hamiltonian? Having seen how to derive Hamilton’s equations on (Z, Ω) given H, it is natural to consider the converse: when is a given set of equations dz = X(z), where X : Z → Z is a vector field, (2.5.1) dt Hamilton’s equations for some H? If X is linear, the answer is given by the following. ........................... 15 July 1998—18h02 ...........................

91. 2.5 When Are Equations Hamiltonian? 77 Proposition 2.5.1. Let the vector field A : Z → Z be linear. Then A is Hamiltonian if and only if A is Ω-skew; that is, Ω(Az1 , z2 ) = −Ω(z1 , Az2 ) for all z1 , z2 ∈ Z. Furthermore, in this case one can take H(z) = 1 Ω(Az, z). 2 Proof. Differentiating the defining relation Ω(XH (z), v) = dH(z) · v (2.5.2) with respect to z in the direction u and using bilinearity of Ω, one gets Ω(DXH (z) · u, v) = D2 H(z)(v, u). (2.5.3) From this and the symmetry of the second partial derivatives, we get Ω(DXH (z) · u, v) = D2 H(z)(u, v) = Ω(DXH (z) · v, u) = −Ω(u, DXH (z) · v). (2.5.4) If A = XH for some H, then DXH (z) = A, and (2.5.4) becomes Ω(Au, v) = −Ω(u, Av); hence A is Ω-skew. Conversely, suppose that A is Ω-skew. Defining H(z) = 1 Ω(Az, z), we 2 claim that A = XH . Indeed, dH(z) · u = 1 Ω(Au, z) + 1 Ω(Az, u) 2 2 = − 1 Ω(u, Az) + 1 Ω(Az, u) 2 2 = 1 Ω(Az, u) + 1 Ω(Az, u) = Ω(Az, u). 2 2 In canonical coordinates, where Ω has matrix J, Ω-skewness of A is equivalent to symmetry of the matrix JA; that is, JA + AT J = 0. The vector space of all linear transformations of Z satisfying this condition is denoted by sp(Z, Ω) and its elements are called infinitesimal symplectic transformations. In canonical coordinates, if Z = W × W ∗ and if A has the matrix Aqq Aqp A= , (2.5.5) Apq App then one checks that A is infinitesimally symplectic if and only if Aqp and Apq are both symmetric and AT + App = 0 (see Exercise 2.5-1). qq In the complex linear case, we use Example (f) in §2.2 (2 times the negative imaginary part of a Hermitian inner product , is the symplectic form) to arrive at the following. ........................... 15 July 1998—18h02 ...........................

92. 78 2. Hamiltonian Systems on Linear Symplectic Spaces Corollary 2.5.2. Let H be a complex Hilbert space with Hermitian inner product , and let Ω(ψ1 , ψ2 ) = −2 Im ψ1 , ψ2 . Let A : H → H be a complex linear operator. There exists an H : H → R such that A = XH if and only if iA is symmetric or, equivalently, satisfies iAψ1 , ψ2 = ψ1 , iAψ2 . (2.5.6) In this case, H may be taken to be H(ψ) = iAψ, ψ . We let Hop = ˙ i A and thus Hamilton’s equations ψ = Aψ becomes the Schr¨dinger o equation 3 : ∂ψ i = Hop ψ. (2.5.7) ∂t Proof. A is Ω-skew if and only if Im Aψ1 , ψ2 = − Im ψ1 , Aψ2 for all ψ1 , ψ2 ∈ H. Replacing everywhere ψ1 by iψ1 and using the relation Im(iz) = Re z, this is equivalent to Re Aψ1 , ψ2 = − Re ψ1 , Aψ2 . Since iAψ1 , ψ2 = − Im Aψ1 , ψ2 + i Re Aψ1 , ψ2 , (2.5.8) and ψ1 , iAψ2 = + Im ψ1 , Aψ2 − i Re ψ1 , Aψ2 , (2.5.9) we see that Ω-skewness of A is equivalent to iA being symmetric. Finally 1 iAψ, ψ = Re i Aψ, ψ = − Im Aψ, ψ = Ω(Aψ, ψ) 2 and the corollary follows from Proposition 2.5.1. For nonlinear differential equations, the analog of Proposition 2.5.1 is the following. Proposition 2.5.3. Let X : Z → Z be a (smooth) vector field on a symplectic vector space (Z, Ω). Then X = XH for some H : Z → R if and only if DX(z) is Ω-skew for all z. Proof. We have seen the “only if” part in the proof of Proposition 2.5.1. Conversely, if DX(z) is Ω-skew, define4 1 H(z) = Ω(X(tz), z) dt + constant; (2.5.10) 0 3 Strictly speaking, equation (2.5.6) is required to hold only on the domain of the operator A, which need not be all of H. We shall ignore these issues for simplicity. This example is continued in §2.6 and in §3.2. 4 Looking ahead to Chapter 4 on differential forms, one can check that (2.5.10) for H is reproduced by the proof of the Poincar´ lemma applied to the one-form iX Ω. That e DX(z) is Ω-skew is equivalent to d(iX Ω) = 0. ........................... 15 July 1998—18h02 ...........................

93. 2.5 When Are Equations Hamiltonian? 79 we claim that X = XH . Indeed, 1 dH(z) · v = [Ω(DX(tz) · tv, z) + Ω(X(tz), v)] dt 0 1 = [Ω(tDX(tz) · z, v) + Ω(X(tz), v)] dt 0 1 =Ω [tDX(tz) · z + X(tz)] dt, v 0 1 d =Ω [tX(tz)] dt, v = Ω(X(z), v). 0 dt This is out Using the straightening out theorem (see, for example, Abraham, Mars- of place– this requires den, and Ratiu [1988], Section 4.1) it is easy to see that on an even- symplectic dimensional manifold any vector field is locally Hamiltonian near points forms–we are where it is non-zero, relative to some symplectic form. However, it is not still in vector so simple to get a general criterion of this sort that is global, covering spaces singular points as well. An interesting characterization of Hamiltonian vector fields involves the Cayley transform. Let (Z, Ω) be a symplectic vector space and A : Z → Z a linear transformation such that I −A is invertible. Then A is Hamiltonian if and only if its Cayley transform C = (I +A)(I −A)−1 is symplectic. See Exercise 2.5-2. For applications, see Laub and Meyer [1974], Paneitz [1981], Feng [1986], and Austin and Krishnaprasad [1993]. The Cayley transform is useful in some Hamiltonian numerical algorithms, as this last reference and Marsden [1992] shows. Exercises 2.5-1. Let Z = W × W ∗ and use a canonical basis to write the matrix of the linear map A : Z → Z as Aqq Aqp A= . Apq App Show that A is infinitesimally symplectic, that is, JA + AT J = 0 if and only if Aqp and Apq are both symmetric and AT + App = 0. qq 2.5-2. Let (Z, Ω) be a symplectic vector space. Let A : Z → Z be a linear map and assume that (I − A) is invertible. Show that A is Hamiltonian if and only if its Cayley transform (I + A)(I − A)−1 is symplectic. Give an example of a linear Hamiltonian vector field such that (I − A) is not invertible. ........................... 15 July 1998—18h02 ...........................

94. 80 2. Hamiltonian Systems on Linear Symplectic Spaces 2.5-3. Suppose that (Z, Ω) is a finite-dimensional symplectic vector space and let ϕ : Z → Z be a linear symplectic map. If λ is an eigenvalue of multiplicity k, then so is 1/λ. Prove this using the characteristic polynomial of ϕ. 2.5-4. Suppose that (Z, Ω) is a finite-dimensional symplectic vector space and let A : Z → Z be a Hamiltonian vector field. Show that the generalized kernel of A defined to be the set {z ∈ Z | Ak z = 0, for some integer k ≥ 1}, is a symplectic subspace. 2.6 Hamiltonian Flows This subsection discusses flows of Hamiltonian vector fields a little further. The next subsection gives the abstract definition of the Poisson bracket, relates it to the classical definitions, and then shows how it may be used in describing the dynamics. Later on, Poisson brackets will play an increas- ingly important role. Let XH be a Hamiltonian vector field on a symplectic vector space (Z, Ω) with Hamiltonian H : Z → R. The flow of XH is the collection of maps ϕt : Z → Z satisfying d ϕt (z) = XH (ϕt (z)) (2.6.1) dt for each z ∈ Z and real t. Here and in the following, all statements concerning the map ϕt : Z → Z are to be considered only for those z and t such that ϕt (z) is defined, as determined by differential equations theory. Linear Flows. First consider the case in which A is a (bounded) linear vector field. The flow of A may be written as ϕt = etA ; that is, the solution of dz/dt = Az with initial condition z0 is given by z(t) = ϕt (z0 ) = etA z0 . Proposition 2.6.1. The flow ϕt of a linear vector field A : Z → Z consists of (linear) canonical transformations if and only if A is Hamiltonian. Proof. For all u, v ∈ Z we have d ∗ d (ϕt Ω)(u, v) = Ω(ϕt (u), ϕt (v)) dt dt d d =Ω ϕt (u), ϕt (v) + Ω ϕt (u), ϕt (v) dt dt = Ω(Aϕt (u), ϕt (v)) + Ω(ϕt (u), Aϕt (v)). Therefore, A is Ω-skew, that is, A is Hamiltonian, if and only if each ϕt is a linear canonical transformation. ........................... 15 July 1998—18h02 ...........................

95. 2.6 Hamiltonian Flows 81 Nonlinear Flows. For nonlinear flows, there is a corresponding result. Proposition 2.6.2. The flow ϕt of a (nonlinear) Hamiltonian vector field XH consists of canonical transformations. Conversely, if the flow of a vector field X consists of canonical transformations, then it is Hamiltonian. Proof. Let ϕt be the flow of a vector field X. By (2.6.1) and the chain rule: d d [Dϕt (z) · v] = D ϕt (z) · v = DX(ϕt (z)) · (Dϕt (z) · v). dt dt which is called the first variation equation. Using this, we get d Ω(Dϕt (z) · u, Dϕt (z) · v) = Ω(DX(ϕt (z)) · [Dϕt (z) · u], Dϕt (z) · v) dt + Ω(Dϕt (z) · u, DX(ϕt (z)) · [Dϕt (z) · v]). If X = XH , then DXH (ϕt (z)) is Ω-skew by Proposition 2.5.3, so, Ω(Dϕt (z) · u, Dϕt (z) · v) = constant. At t = 0 this equals Ω(u, v), so ϕ∗ Ω = Ω. Conversely, if ϕt is canonical, this t calculation shows that DX(ϕt (z)) is Ω-skew, whence by Proposition 2.5.3, X = XH for some H. Later on we give another proof of Proposition 2.6.2 using differential forms. Example: Schr¨dinger Equation o Recall that if H is a complex Hilbert space, a complex linear map U : H → H is called unitary if it preserves the Hermitian inner product. Proposition 2.6.3. Let A : H → H be a complex linear map on a complex Hilbert space H. The flow ϕt of A is canonical, that is, consists of canonical transformations with respect to the symplectic form Ω defined in Example (f ) of §2.2, if and only if ϕt is unitary. Proof. By definition, Ω(ψ1 , ψ2 ) = −2 Im ψ1 , ψ2 , so Ω(ϕt ψ1 , ϕt ψ2 ) = −2 Im ϕt ψ1 , ϕt ψ2 for ψ1 , ψ2 ∈ H. Thus ϕt is canonical if and only if Im ϕt ψ1 , ϕt ψ2 = Im ψ1 , ψ2 and this in turn is equivalent to unitarity by complex linearity of ϕt since ψ1 , ψ2 = − Im iψ1 , ψ2 + i Im ψ1 , ψ2 . ........................... 15 July 1998—18h02 ...........................

96. 82 2. Hamiltonian Systems on Linear Symplectic Spaces ˙ This shows that the flow of the Schr¨dinger equation ψ = Aψ is o canonical and unitary and so preserves the probability amplitude of any wave function that is a solution: ϕt ψ, ϕt ψ = ψ, ψ , where ϕt is the flow of A. Later we shall see how this conservation of the norm also results from a symmetry-induced conservation law. 2.7 Poisson Brackets Definition 2.7.1. Given a symplectic vector space (Z, Ω) and two functions F, G : Z → R, the Poisson bracket {F, G} : Z → R of F and G is defined by {F, G}(z) = Ω(XF (z), XG (z)). (2.7.1) Using the definition of a Hamiltonian vector field, we find that equivalent expressions are {F, G}(z) = dF (z) · XG (z) = −dG(z) · XF (z). (2.7.2) In (2.7.2) we write £XG F = dF · XG , for the derivative of F in the direction XG . Lie Derivative Notation. The Lie derivative of f along X, £X f = df ·X is the directional derivative of f in the direction X. In coordinates it is given by ∂f I £X f = X (sum on I). ∂z I Functions F, G which are such that {F, G} = 0 are said to be in involution or to Poisson commute. Examples Now we turn to some examples of Poisson brackets. (a) Canonical Bracket. Suppose that Z is 2n-dimensional. Then in canonical coordinates (q 1 , . . . , q n , p1 , . . . , pn ) we have  ∂G  ∂F ∂F  ∂pi  {F, G} = ,− i J  ∂G   ∂pi ∂q − i ∂q ∂F ∂G ∂F ∂G = − (sum on i). (2.7.3) ∂q i ∂pi ∂pi ∂q i ........................... 15 July 1998—18h02 ...........................

97. 2.7 Poisson Brackets 83 From this, we get the fundamental Poisson brackets: {q i , q j } = 0, {pi , pj } = 0, and {q i , pj } = δj . i (2.7.4) In terms of the Poisson structure, that is, the bilinear form B from §2.1, the Poisson bracket takes the form {F, G} = B(dF, dG). (2.7.5) (b) The Space of Functions. Let equations (Z, Ω) be defined as in Ex- ample (b) of §2.2 and let F, G : Z → R. Using equations (2.4.10) and (2.7.1) above, we get δF δF δG δG {F, G} = Ω(XF , XG ) = Ω ,− , ,− δπ δϕ δπ δϕ δG δF δF δG = − d3 x. (2.7.6) R3 δπ δϕ δπ δϕ This example will be used in the next chapter when we study classical field theory. The Jacobi–Lie Bracket. The Jacobi–Lie bracket [X, Y ] of two vector fields X and Y on a vector space Z is defined by demanding that df · [X, Y ] = d(df · Y ) · X − d(df · X) · Y for all real-valued functions f . In Lie derivative notation, this reads £[X,Y ] f = £X £Y f − £Y £X f. One checks that this condition becomes, in vector analysis notation, [X, Y ] = (X · )Y − (Y · )X, and in coordinates, ∂ J ∂ Y − Y I I XJ . [X, Y ]J = X I I ∂z ∂z Proposition 2.7.2. Let [ , ] denote the Jacobi–Lie bracket of vector fields, and let F, G ∈ F(Z). Then X{F,G} = −[XF , XG ]. (2.7.7) Proof. We calculate as follows: Ω(X{F,G} (z), u) = d{F, G}(z) · u = d(Ω(XF (z), XG (z))) · u = Ω(DXF (z) · u, XG (z)) + Ω(XF (z), DXG (z) · u) = Ω(DXF (z) · XG (z), u) − Ω(DXG (z) · XF (z), u) = Ω(DXF (z) · XG (z) − DXG (z) · XF (z), u) = Ω(−[XF , XG ](z), u). Weak nondegeneracy of Ω implies the result. ........................... 15 July 1998—18h02 ...........................

98. 84 2. Hamiltonian Systems on Linear Symplectic Spaces Jacobi’s Identity. We are now ready to prove the Jacobi identity in a fairly general context. Proposition 2.7.3. Let (Z, Ω) be a symplectic vector space. Then the Poisson bracket { , } : F(Z) × F(Z) → F(Z) makes F(Z) into a Lie algebra. That is, this bracket is real bilinear, skew-symmetric, and satisfies Jacobi’s identity , that is, {F, {G, H}} + {G, {H, F }} + {H, {F, G}} = 0. Proof. To verify Jacobi’s identity note that for F, G, H : Z → R, we have {F, {G, H}} = −£XF {G, H} = £XF £XG H, {G, {H, F }} = −£XG {H, F } = −£XG £XF H and {H, {F, G}} = £X{F,G} H, so that {F, {G, H}} + {G, {H, F }} + {H, {F, G}} = £X{F,G} H + £[XF ,XG ] H. The result thus follows by (2.7.7). From Proposition 2.7.2 we see that the Jacobi–Lie bracket of two Hamil- tonian vector fields is again Hamiltonian. Thus, we obtain: Corollary 2.7.4. The set of Hamiltonian vector fields XHam (Z) forms a Lie subalgebra of X(Z). Next, we characterize symplectic maps in terms of brackets. Proposition 2.7.5. Let ϕ : Z → Z be a diffeomorphism. Then ϕ is symplectic if and only if it preserves Poisson brackets, that is, {ϕ∗ F, ϕ∗ G} = ϕ∗ {F, G}, (2.7.8) for all F, G : Z → R. Proof. We use the identity ϕ∗ (£X f ) = £ϕ∗ X (ϕ∗ f ), which follows from the chain rule. Thus, ϕ∗ {F, G} = ϕ∗ £XG F = £ϕ∗ XG (ϕ∗ F ) and {ϕ∗ F, ϕ∗ G} = £XG◦ϕ (ϕ∗ F ). Thus ϕ preserves Poisson brackets if and only if ϕ∗ XG = XG◦ϕ for every G : Z → R, that is, if and only if ϕ is symplectic by Proposition 2.4.2. ........................... 15 July 1998—18h02 ...........................

99. 2.8 A Particle in a Rotating Hoop 85 Proposition 2.7.6. Let XH be a Hamiltonian vector field on Z, with Hamiltonian H and flow ϕt . Then for F : Z → R, d (F ◦ ϕt ) = {F ◦ ϕt , H} = {F, H} ◦ ϕt . (2.7.9) dt Proof. By the chain rule and the definition of XF , d [(F ◦ ϕt )(z)] = dF (ϕt (z)) · XH (ϕt (z)) dt = Ω(XF (ϕt (z)), XH (ϕt (z))) = {F, H}(ϕt (z)). By Proposition 2.6.2 and (2.7.8), this equals {F ◦ ϕt , H ◦ ϕt }(z) = {F ◦ ϕt , H}(z) by conservation of energy. Corollary 2.7.7. Let F, G : Z → R. Then F is constant along integral curves of XG if and only if G is constant along integral curves of XF and this is true if and only if {F, G} = 0. Proposition 2.7.8. Let A, B : Z → Z be linear Hamiltonian vector fields with corresponding energy functions HA (z) = 1 Ω(Az, z) 2 and HB (z) = 1 Ω(Bz, z). 2 Letting [A, B] = A ◦ B − B ◦ A be the operator commutator, we have {HA , HB } = H[A,B] . (2.7.10) Proof. By definition, XHA = A and so {HA , HB }(z) = Ω(Az, Bz). Since A and B are Ω-skew, we get {HA , HB }(z) = 1 Ω(ABz, z) − 1 Ω(BAz, z) 2 2 = 1 Ω([A, B]z, z) = H[A,B] (z). 2 2.8 A Particle in a Rotating Hoop In this subsection we take a break from the abstract theory to do an example the “old-fashioned” way. This and other examples will also serve as excellent illustrations of the theory we are developing. ........................... 15 July 1998—18h02 ...........................

100. 86 2. Hamiltonian Systems on Linear Symplectic Spaces z ω R eθ ϕ y θ eϕ x er Figure 2.8.1. A particle moving in a hoop rotating with angular velocity ω. Derivation of the Equations. Consider a particle constrained to move on a circular hoop; for example a bead sliding in a hula-hoop. The particle is assumed to have mass m and to be acted on by gravitational and frictional forces, as well as constraint forces that keep it on the hoop. The hoop itself is spun about a vertical axis with constant angular velocity ω, as in Figure 2.8.1. The position of the particle in space is speciﬁed by the angles θ and ϕ, as shown in Figure 2.8.1. We can take ϕ = ωt, so the position of the particle becomes determined by θ alone. Let the orthonormal frame along the coordinate directions eθ , eϕ , and er be as shown. The forces acting on the particle are: 1. Friction, proportional to the velocity of the particle relative to the ˙ hoop: −νRθeθ , where ν ≥ 0 is a constant. 2. Gravity: −mgk. 3. Constraint forces in the directions er and eϕ to keep the particle in the hoop. The equations of motion are derived from Newton’s second law F = ma. To get them, we need to calculate the acceleration a; here a means the acceleration relative to the ﬁxed inertial frame xyz in space; it does not ........................... 15 July 1998—18h02 ...........................

101. 2.8 A Particle in a Rotating Hoop 87 ¨ mean θ. Relative to this xyz coordinate system, we have x = R sin θ cos ϕ, y = R sin θ sin ϕ, (2.8.1) z = −R cos θ. Calculating the second derivatives using ϕ = ωt and the chain rule gives ˙ ¨ ˙ x = −ω 2 x − θ2 x + (R cos θ cos ϕ)θ − 2Rω θ cos θ sin ϕ, ¨ ˙ ¨ ˙ y = −ω 2 y − θ2 y + (R cos θ sin ϕ)θ + 2Rω θ cos θ cos ϕ, ¨ (2.8.2) ˙2 ¨ z = −z θ + (R sin θ)θ. ¨ If i, j, k, denote unit vectors along the x, y, and z axes, respectively, we have the easily verified relation eθ = (cos θ cos ϕ)i + (cos θ sin ϕ)j + sin θk. (2.8.3) Now consider the vector equation F = ma, where F is the sum of the three forces described earlier and a = xi + y j + z k. ¨ ¨ ¨ (2.8.4) The eϕ and er components of F = ma only tell us what the constraint forces must be; the equation of motion comes from the eθ component: F · eθ = ma · eθ . (2.8.5) Using (2.8.3), the left side of (2.8.5) is ˙ F · eθ = −νRθ − mg sin θ (2.8.6) while from (2.8.2), (2.8.3), and (2.8.4), the right side of (2.8.5) is ma · eθ = m{¨ cos θ cos ϕ + y cos θ sin ϕ + z sin θ} x ¨ ¨ = m{cos θ cos ϕ[−ω x − θ 2 ˙2 x + (R cos θ cos ϕ)θ ¨ ˙ ˙ − 2Rω θ cos θ sin ϕ] + cos θ sin ϕ[−ω 2 y − θ2 y ¨ ˙ + (R cos θ sin ϕ)θ + 2Rω θ cos θ cos ϕ] ˙ ¨ + sin θ[−z θ2 + (R sin θ)θ]}. Using (2.8.1), this simplifies to ¨ ma · eθ = mR{θ − ω 2 sin θ cos θ}. (2.8.7) Comparing (2.8.5), (2.8.6), and (2.8.7), we get ¨ ν ˙ g θ = ω 2 sin θ cos θ − θ − sin θ (2.8.8) m R as our final equation of motion. Several remarks concerning it are in order: ........................... 15 July 1998—18h02 ...........................

102. 88 2. Hamiltonian Systems on Linear Symplectic Spaces (i) If ω = 0 and ν = 0, (2.8.8) reduces to the pendulum equation ¨ Rθ + g sin θ = 0. In fact, our system can be viewed just as well as a whirling pendulum. ˙ (ii) For ν = 0, (2.8.8) is Hamiltonian with respect to q = θ, p = mR2 θ, canonical bracket structure ∂F ∂K ∂K ∂F {F, K} = − , (2.8.9) ∂q ∂p ∂q ∂p and the Hamiltonian p2 mR2 ω 2 H= − mgR cos θ − sin2 θ. (2.8.10) 2mR2 2 Derivation as Euler–Lagrange Equations. We now use Lagrangian methods to derive (2.8.8). In Figure 2.8.1, the velocity is ˙ v = Rθeθ + (ωR sin θ)eϕ , so the kinetic energy is T = 1m v 2 ˙ = 1 m(R2 θ2 + [ωR sin θ]2 ), (2.8.11) 2 2 while the potential energy is V = −mgR cos θ. (2.8.12) Thus the Lagrangian is given by 1 ˙ mR2 ω 2 L=T −V = mR2 θ2 + sin2 θ + mgR cos θ (2.8.13) 2 2 and the Euler–Lagrange equations, namely, d ∂L ∂L = , ˙ dt ∂ θ ∂θ (see §1.1 or §2.1) become ¨ mR2 θ = mR2 ω 2 sin θ cos θ − mgR sin θ, which are the same equations we derived by hand in (2.8.8) for ν = 0. The ˙ Legendre transform gives p = mR2 θ and the Hamiltonian (2.8.10). Notice that this Hamiltonian is not the kinectic plus potential energy of the particle. In fact, if one postulated this, then Hamilton’s equations would give the incorrect equations. This has to do with deeper covariance properties of the Lagrangian versus Hamiltonian equations. ........................... 15 July 1998—18h02 ...........................

103. 2.8 A Particle in a Rotating Hoop 89 ˙ Equilibria. The equilibrium solutions are solutions satisfying θ = 0, ¨ = 0; (2.8.8) gives θ Rω 2 sin θ cos θ = g sin θ. (2.8.14) Certainly, θ = 0 and θ = π solve (2.8.14) corresponding to the particle at the bottom or top of the hoop. If θ = 0 or π, (2.8.14) becomes Rω 2 cos θ = g (2.8.15) which has two solutions when g/Rω 2 < 1. The value g ωc = (2.8.16) R is the critical rotation rate. (Notice that ωc is the frequency of linearized ¨ oscillations for the simple pendulum, that is, for Rθ + gθ = 0.) For ω < ωc there are only two solutions θ = 0, π, while for ω > ωc there are four solutions, g θ = 0, π, ± cos−1 . (2.8.17) Rω 2 We say that a bifurcation (or a Hamiltonian pitchfork bifurcation to be accurate) has occurred as ω crosses ωc . We can see this graphically ˙ in computer generated solutions of (2.8.8). Set x = θ, y = θ and rewrite (2.8.8) as x = y, ˙ g (2.8.18) y = (α cos x − 1) sin x − βy, ˙ R where α = Rω 2 /g and β = ν/m. Taking g = R for illustration, Figure 2.8.2 shows representative orbits in the phase portraits of (2.8.18) for various α, β. This system with ν = 0; that is, β = 0, is symmetric in the sense that the ˙ ˙ Z2 -action given by θ → −θ and θ → −θ leaves the phase portrait invariant. If this Z2 symmetry is broken, by setting the rotation axis a little off center, for example, then one side gets preferred, as in Figure 2.8.3. The evolution of the phase portrait for ν = 0 is shown in Figure 2.8.4. Near θ = 0, the potential function has changed from the symmetric bifurcation in Figure 2.8.5(a) to the unsymmetric one in Figure 2.8.5(b). This is what is known as the cusp catastrophe; see Golubitsky and Schaeffer [1985] and Arnold [1968, 1984] for more information. In (2.8.8), imagine that the hoop is subject to small periodic pulses; say ω = ω0 + ρ cos(ηt). Using the Melnikov method described in the introduction and in the following section, it is presumably true (but a messy calculation to prove) that the resulting time-periodic system has horseshoe chaos if and ν are small (where measures how off-center the hoop is), but ρ/ν exceeds a critical value. See Exercise 2.8-3 and §2.11. ........................... 15 July 1998—18h02 ...........................

104. 90 2. Hamiltonian Systems on Linear Symplectic Spaces . . θ θ θ θ . α = 0.5, β = 0 θ α = 1.5, β = 0 θ α = 1.5, β = 0.1 Figure 2.8.2. Phase portraits of the ball in the rotating hoop. ω Figure 2.8.3. A ball in an oﬀ-center rotating hoop. Exercises 2.8-1. Derive the equations of motion for a particle in a hoop spinning about a line a distance oﬀ center. What can you say about the equilibria as functions of and ω? 2.8-2. Derive the formula of Exercise 1.9-1 for the homoclinic orbit (the ¨ orbit tending to the saddle point as t → ±∞) of a pendulum ψ + sin ψ = 0. Do this using conservation of energy, determining the value of the energy ˙ on the homoclinic orbit, solving for ψ and then integrating. ........................... 15 July 1998—18h02 ...........................

105. 2.8 A Particle in a Rotating Hoop 91 Figure 2.8.4. The phase portraits for the ball in the off-centered hoop as the angular velocity increases. (a) ε = 0 (b) ε > 0 Figure 2.8.5. The evolution of the potential for the ball in the centered and the off-centered hoop. 2.8-3. Using the method of the preceding exercise, derive an integral formula for the homoclinic orbit of the frictionless particle in a rotating hoop. 2.8-4. Determine all equilibria of Duffing’s equation x − βx + αx3 = 0, ¨ where α and β are positive constants and study their stability. Derive a formula for the two homoclinic orbits. 2.8-5. Determine the equations of motion and bifurcations for a ball in a light rotating hoop, but this time the hoop is not forced to rotate with constant angular velocity, but rather is free to rotate so that its angular momentum µ is conserved. 2.8-6. Consider the pendulum shown in Figure 2.8.6. It is a planar pendulum whose suspension point is being whirled in a circle with angular velocity ω, by means of a vertical shaft, as shown. The plane of the pendulum is orthogonal to the radial arm of length R. Ignore frictional effects. ........................... 15 July 1998—18h02 ...........................

106. 92 2. Hamiltonian Systems on Linear Symplectic Spaces R l = pendulum length ω m = pendulum bob mass shaft g = gravitational acceleration l R = radius of circle g ω = angular velocity of shaft θ = angle of pendulum from θ the downward vertical m Figure 2.8.6. A whirling pendulum. (i) Using the notation in the figure, find the equations of motion of the pendulum. (ii) Regarding ω as a parameter, show that a supercritical pitchfork bifurcation of equilibria occurs as the angular velocity of the shaft is increased. 2.9 The Poincar´–Melnikov Method and e Chaos Recall from the introduction that in the simplest version of the Poincar´– e Melnikov method we are concerned with dynamical equations that perturb a planar Hamiltonian system z = X0 (z) ˙ (2.9.1) to one of the form z = X0 (z) + X1 (z, t), ˙ (2.9.2) where is a small parameter, z ∈ R2 , X0 is a Hamiltonian vector field with energy H0 , X1 is periodic with period T , and is Hamiltonian with energy a T -periodic function H1 . We assume that X0 has a homoclinic orbit z(t) so z(t) → z0 , a hyperbolic saddle point, as t → ±∞. Define the Poincar´–Melnikov function by e ∞ M (t0 ) = {H0 , H1 }(z(t − t0 ), t) dt (2.9.3) −∞ where { , } denotes the Poisson bracket. There are two convenient ways of visualizing the dynamics of (2.9.2). Introduce the Poincar´ map P s : R2 → R2 , which is the time T map for e ........................... 15 July 1998—18h02 ...........................

107. 2.9 The Poincar´–Melnikov Method and Chaos e 93 (2.9.2) starting at time s. For = 0, the point z0 and the homoclinic orbit s are invariant under P0 , which is independent of s. The hyperbolic saddle z0 persists as a nearby family of saddles z for > 0, small, and we are interested in whether or not the stable and unstable manifolds of the point z for the map P s intersect transversally (if this holds for one s, it holds for all s). If so, we say (2.9.2) has horseshoes for > 0. The second way to study (2.9.2) is to look directly at the suspended system on R2 × S 1 , where S 1 is the circle; (2.9.2) becomes the autonomous suspended system z = X0 (z) + X1 (z, θ), ˙ (2.9.4) ˙ θ = 1. From this point of view, θ gets identified with time and the curve γ0 (t) = (z0 , t) is a periodic orbit for (2.9.4). This orbit has stable manifolds and unsta- s u ble manifolds denoted W0 (γ0 ) and W0 (γ0 ) defined as the set of points tending exponentially to γ0 as t → ∞ and t → −∞, respectively. (See Abraham, Marsden, and Ratiu [1988], Guckenheimer and Holmes [1983], or Wiggins [1988, 1990, 1992] for more details.) In this example, they coincide: s u W0 (γ0 ) = W0 (γ0 ). For > 0 the (hyperbolic) closed orbit γ0 perturbs to a nearby (hyperbolic) closed orbit which has stable and unstable manifolds W s (γ ) and W u (γ ). If W s (γ ) and W u (γ ) intersect transversally, we again say that (2.9.2) has horseshoes. These two definitions of admitting horseshoes are readily seen to be equivalent. Theorem 2.9.1 (Poincar´–Melnikov Theorem). Let the Poincar´– e e Melnikov function be defined by (2.9.3). Assume M (t0 ) has simple zeros as a T -periodic function of t0 . Then, for sufficiently small , (2.9.2) has horseshoes; that is, homoclinic chaos in the sense of transversal intersecting separatrices. Idea of the Proof. In the suspended picture, we use the energy function H0 to measure the first-order movement of W s (γ ) at z(0) at time t0 as is varied. Note that points of z(t) are regular points for H0 since H0 is constant on z(t) and z(0) is not a fixed point. That is, the differential of H0 does not vanish at z(0). Thus, the values of H0 give an accurate measure of the distance from the homoclinic orbit. If (z s (t, t0 ), t) is the curve on W s (γ ) that is an integral curve of the suspended system and has an initial condition z s (t0 , t0 ) that is the perturbation of W0 (γ0 ) ∩ {the plane t = t0 } s ........................... 15 July 1998—18h02 ...........................

108. 94 2. Hamiltonian Systems on Linear Symplectic Spaces in the normal direction to the homoclinic orbit, then H0 (z s (t0 , t0 )) measures the normal distance. But H0 (z s (τ+ , t0 )) − H0 (z s (t0 , t0 )) τ+ d = H0 (z s (t, t0 )) dt t0 dt τ+ = {H0 , H0 + H1 }(z s (t, t0 ), t) dt. (2.9.5) t0 From invariant manifold theory one learns that z s (t, t0 ) converges exponentially to γ (t), a periodic orbit for the perturbed system as t → +∞. Notice from the right hand side of the first equality above that if z s (t, t0 ) is replaced by the periodic orbit γ (t), the result would be zero. Since the convergence is exponential, one concludes that the integral is of order for an interval from some large time to infinity. To handle the finite portion of the integral, we use the fact that z s (t, t0 ) is -close to z(t − t0 ) (uniformly as t → +∞), and that {H0 , H0 } = 0. Therefore, we see that {H0 , H0 + H1 }(z s (t, t0 ), t) = {H0 , H1 }(z(t − t0 ), t) + O( 2 ). Using this over a large but finite interval [t0 , t1 ] and the exponential close- ness over the remaining interval [t1 , ∞), we see that (2.9.5) becomes H0 (z s (τ+ , t0 )) − H0 (z s (t0 , t0 )) τ+ = {H0 , H1 }(z(t − t0 ), t) dt + O( 2 ), (2.9.6) t0 where the error is uniformly small as τ+ → ∞. Similarly, H0 (z u (t0 , t0 )) − H0 (z u (τ− , t0 )) t0 = {H0 , H1 }(z(t − t0 ), t) dt + O( 2 ). (2.9.7) τ− Again we use the fact that z s (τ+ , t0 ) → γ (τ+ ) exponentially fast, a periodic orbit for the perturbed system as τ+ → +∞. Notice that since the orbit is homoclinic, the same periodic orbit can be used for negative times as well. Using this observation, we can choose τ+ and τ− such that H0 (z s (τ+ , t0 ))−H0 (z u (τ− , t0 )) → 0 as τ+ → ∞, τ− → −∞. Thus, adding (2.9.6) and (2.9.7), and letting τ+ → ∞, τ− → −∞, we get H0 (z u (t0 , t0 )) − H0 (z s (t0 , t0 )) ∞ = {H0 , H1 }(z(t − t0 ), t) dt + O( 2 ). (2.9.8) −∞ ........................... 15 July 1998—18h02 ...........................

109. 2.9 The Poincar´–Melnikov Method and Chaos e 95 The integral in this expression is convergent because the curve z(t − t0 ) tends exponentially to the saddle point as t → ±∞, and because the differential of H0 vanishes at this point. Thus, the integrand tends to zero exponentially fast as t tends to plus and minus infinity. Since the energy is a “good” measure of the distance between the points z u (t0 , t0 )) and z s (t0 , t0 )), it follows that if M (t0 ) has a simple zero at time t0 , then z u (t0 , t0 ) and z s (t0 , t0 ) intersect transversally near the point z(0) at time t0 . If in (2.9.2), only X0 is Hamiltonian, the same conclusion holds if (2.9.3) is replaced by ∞ M (t0 ) = (X0 × X1 )(z(t − t0 ), t) dt, (2.9.9) −∞ where X0 × X1 is the (scalar) cross product for planar vector fields. In fact, X0 need not even be Hamiltonian if an area expansion factor is inserted. Example A. Equation (2.9.9) applies to the forced damped Duffing equation u − βu + αu3 = (γ cos ωt − δ u). ¨ ˙ (2.9.10) Here the homoclinic orbits are given by (see Exercise 2.8-4) 2β u(t) = ± sech( βt) (2.9.11) α and (2.9.9) becomes, after a residue calculation, 2 πω 4δβ 3/2 M (t0 ) = γπω sech √ sin(ωt0 ) − , (2.9.12) α 2 β 3α so one has simple zeros and hence chaos of the horseshoe type if √ γ 2 2β 3/2 πω > √ cosh √ (2.9.13) δ 3ω α 2 β and is small. Example B. Another interesting example, due to Montgomery [1985], concerns the equations for superfluid 3 He. These are the Leggett equations and we shall confine ourselves to what is called the A phase for simplicity (see Montgomery’s paper for additional results). The equations are 1 χΩ2 s=− ˙ sin 2θ 2 γ2 ........................... 15 July 1998—18h02 ...........................

110. 96 2. Hamiltonian Systems on Linear Symplectic Spaces and ˙ γ2 1 θ= s− γB sin ωt + Γ sin 2θ . (2.9.14) χ 2 Here s is the spin, θ an angle (describing the “order parameter”), and γ, χ, . . . are physical constants. The homoclinic orbits for = 0 are given by Ωe±2Ωt θ± = 2 tan−1 (e±Ωt ) − π/2 and s± = ±2 . (2.9.15) 1 + e±2Ωt One calculates the Poincar´–Melnikov function to be e πχωB ωπ 2 χ M± (t0 ) = sech cos ωt − ΩΓ, (2.9.16) 8γ 2Ω 3 γ2 so that (2.9.14) has chaos in the sense of horseshoes if γB 16 Ω πω > cosh (2.9.17) Γ 3π ω 2Ω and if is small. For references and information on higher-dimensional versions of the method and applications, see Wiggins [1988]. We shall comment on some aspects of this shortly. There is even a version of the Poincar´–Melnikov e method applicable to PDEs (due to Holmes and Marsden [1981]). One basi- cally still uses formula (2.9.9) where X0 × X1 is replaced by the symplectic pairing between X0 and X1 . However, there are two new difficulties in addition to standard technical analytic problems that arise with PDEs. The first is that there is a serious problem with resonances. This can be dealt with using the aid of damping. Second, the problem seems to be not re- ducible to two dimensions; the horseshoe involves all the modes. Indeed, the higher modes do seem to be involved in the physical buckling processes for the beam model discussed next. Example C. A PDE model for a buckled forced beam is 1 w + w + Γw − κ ¨ [w ]2 dz w = (f cos ωt − δ w), ˙ (2.9.18) 0 where w(z, t), 0 ≤ z ≤ 1, describes the deflection of the beam, · = ∂/∂t, = ∂/∂z, and Γ, κ, . . . are physical constants. For this case, one finds that if (i) π 2 < Γ < 4ρ3 (first mode is buckled); ........................... 15 July 1998—18h02 ...........................

111. 2.9 The Poincar´–Melnikov Method and Chaos e 97 (ii) j 2 π 2 (j 2 π 2 − Γ) = ω 2 , j = 2, 3, . . . (resonance condition); f π(Γ − π 2 ) ω (iii) > √ cosh √ (transversal zeros for M (t0 )); δ 2ω κ 2 Γ − ω2 (iv) δ > 0; and is small, then (2.9.18) has horseshoes. Experiments (see Moon [1988]) showing chaos in a forced buckled beam provided the motivation which lead to the study of (2.9.18). This kind of result can also be used for a study of chaos in a van der Waals fluid (Slemrod and Marsden [1985]) and for soliton equations (see Birnir [1986], Ercolani, Forest, and McLaughlin [1990], and Birnir and Grauer [1994]). For example, in the damped, forced sine-Gordon equation one has chaotic transitions between breathers and kink-antikink pairs and in the Benjamin–Ono equation one can have chaotic transitions between solutions with different numbers of poles. More Degrees of Freedom. For Hamiltonian systems with two degrees of freedom, Holmes and Marsden [1982a] show how the Melnikov method may be used to prove the existence of horseshoes on energy surfaces in nearly integrable systems. The class of systems studied have a Hamiltonian of the form H(q, p, θ, I) = F (q, p) + G(I) + H1 (q, p, θ, I) + O( 2 ), (2.9.19) where (θ, I) are action-angle coordinates for the oscillator G; G(0) = 0, G > 0. It is assumed that F has a homoclinic orbit x(t) = (q(t), p(t)) and that ∞ M (t0 ) = {F, H1 } dt, (2.9.20) −∞ the integral taken along (x(t − t0 ), Ωt, I) has simple zeros. Then (2.9.19) has horseshoes on energy surfaces near the surface corresponding to the homoclinic orbit and small I; the horseshoes are taken relative to a Poincar´ e map strobed to the oscillator G. The paper by Holmes and Marsden [1982a] also studies the effect of positive and negative damping. These results are related to those for forced one degree of freedom systems since one can often reduce a two degrees of freedom Hamiltonian system to a one degree of freedom forced system. For some systems in which the variables do not split as in (2.9.19), such as a nearly symmetric heavy top, one needs to exploit a symmetry of the system and this complicates the situation to some extent. The general theory for this is given in Holmes and Marsden [1983] and was applied to show the existence of horseshoes in the nearly symmetric heavy top; see also some closely related results of Ziglin [1980a]. ........................... 15 July 1998—18h02 ...........................

112. 98 2. Hamiltonian Systems on Linear Symplectic Spaces This theory has been used by Ziglin [1980b] and Koiller [1985] in vortex dynamics, for example, to give a proof of the non-integrability of the restricted four vortex problem. Koiller, Soares and Melo Neto [1985] gives applications to the dynamics of general relativity showing the existence of horseshoes in Bianchi IX models. See Oh, Sreenath, Krishnaprasad, and Marsden [1989] for applications to the dynamics of coupled rigid bodies. Arnold [1964] extended the Poincar´–Melnikov theory to systems with e several degrees of freedom. In this case the transverse homoclinic manifolds are based on KAM tori and allow the possibility of chaotic drift from one torus to another. This drift, now known as Arnold diffusion is a much studied ingredient in Hamiltonian systems but its theoretical foundation is still uncertain. Instead of a single Melnikov function, in the multidemnsional case one has a Melnikov vector given schematically by  ∞  −∞ {H0 , H1 } dt  ∞ {I , H } dt   −∞ 1 1  M=  ,  (2.9.21)  ...  ∞ −∞ n {I , H1 } dt where I1 , . . . , In are integrals for the unperturbed (completely integrable) system and where M depends on t0 and on angles conjugate to I1 , . . . , In . One requires M to have transversal zeros in the vector sense. This result was given by Arnold for forced systems and was extended to the autonomous case by Holmes and Marsden [1982b, 1983]; see also Robinson [1988]. These results apply to systems such as a pendulum coupled to several oscillators and the many vortex problems. It has also been used in power systems by Salam, Marsden, and Varaiya [1983], building on the horseshoe case treated by Kopell and Washburn [1982]. See also Salam and Sastry [1985]. There have been a number of other directions of research on these techniques. For example, Grundler [1985] developed a multidimensional version applicable to the spherical pendulum and Greenspan and Holmes [1983] showed how it can be used to study subharmonic bifurcations. See Wiggins [1988] for more information. Poincar´ and Exponentially Small Terms. In Poincar´’s celebrated e e memoir [1890] on the three-body problem, he introduced the mechanism of transversal intersection of separatrices which obstructs the integrability of the equations and the attendant convergence of series expansions for the solutions. This idea has been developed by Birkhoff and Smale using the horseshoe construction to describe the resulting chaotic dynamics. How- ever, in the region of phase space studied by Poincar´, it has never been e proved (except in some generic sense that is not easy to interpret in specific cases) that the equations really are nonintegrable. In fact, Poincar´ him- e self traced the difficulty to the presence of terms in the separatrix splitting ........................... 15 July 1998—18h02 ...........................

113. 2.9 The Poincar´–Melnikov Method and Chaos e 99 which are exponentially small. A crucial component of the measure of the splitting is given by the following formula of Poincar´ [1890, p. 223]: e −8πi J= , exp √π 2µ + exp − √π 2µ which is exponentially small (or beyond all orders) in µ. Poincar´ was aware e of the difficulties that this exponentially small behavior causes; on page 224 of his article, he states: “En d’autres termes, si on regarde µ comme un infiniment petit du premier ordre, la distance BB , sans ˆtre nulle, est e un infiniment petit d’ordre infini. C’est ainsi que la fonction e−1/µ est un infiniment petit d’ordre infini sans ˆtre nulle . . . Dans l’example particulier e que nous avons trait´ plus haut, la distance BB √ du m`me ordre de e est e grandeur que l’integral J, c’est ` dire que exp(−π/ 2µ).” a This is a serious difficulty that arises when one uses the Melnikov method near an elliptic fixed point in a Hamiltonian system or in bifurcation problems giving birth to homoclinic orbits. The difficulty is related to those described by Poincar´. Near elliptic points, one sees homoclinic orbits in e normal forms and after a temporal rescaling this leads to a rapidly os- cillatory perturbation that is modeled by the following variation of the pendulum equation: ¨ ωt φ + sin φ = cos . (2.9.22) If one formally computes M (t0 ) one finds: πω ωt0 M (t0 , ) = ±2π sech cos . (2.9.23) 2 While this has simple zeros, the proof of the Poincar´–Melnikov theorem e is no longer valid since M (t0 , ) is now of order exp(−π/2 ) and the error analysis in the proof only gives errors of order 2 . In fact, no expansion in powers of can detect exponentially small terms like exp(−π/2 ). Holmes, Marsden, and Scheurle [1988] and Delshams and Seara [1991] show that (2.9.22) has chaos that is, in a suitable sense, exponentially small in . The idea is to expand expressions for the stable and unstable manifolds in a Perron type series whose terms are of order k exp(−π/2 ). To do so, the extension of the system to complex time plays a crucial role. One can hope that since such results for (2.9.22) can be proved, it may be possible to return to Poincar´’s 1890 work and complete the arguments he left e unfinished. In fact, these exponentially small phenomena is one reason that the problem of Arnold diffusion is both hard and delicate. To illustrate how exponentially small phenomena enter bifurcation problems, consider the problem of a Hamiltonian saddle node bifurcation x + µx + x2 = 0 ¨ (2.9.24) ........................... 15 July 1998—18h02 ...........................

114. 100 2. Hamiltonian Systems on Linear Symplectic Spaces with the addition of higher-order terms and forcing: x + µx + x2 + h.o.t. = δf (t). ¨ (2.9.25) The phase portrait of (2.9.24) is shown in Figure 2.9.1. . . x x −µ x −µ x µ<0 µ>0 Figure 2.9.1. Phase portraits of x + µx + x2 = 0. ¨ The system (2.9.24) is Hamiltonian with 1 2 1 2 1 3 H(x, x) = ˙ x + µx + x . ˙ (2.9.26) 2 2 3 Let us ﬁrst consider the system without higher-order terms: x + µx + x2 = δf (t). ¨ (2.9.27) To study it, we rescale to blow up the singularity; let x(t) = λξ(τ ), (2.9.28) √ where λ = |µ| and τ = t λ. Letting = d/dτ , we get  δ τ  ξ − ξ + ξ2 = f √ , µ < 0,    µ2 −µ  (2.9.29)   δ τ  2 ξ + ξ + ξ = 2f √ , µ > 0,   µ µ The exponentially small estimates of Holmes, Marsden, and Scheurle [1988] apply to (2.9.29). One gets exponentially small upper and lower estimates in certain algebraic sectors of the (δ, µ) plane that depend on the nature ........................... 15 July 1998—18h02 ...........................

115. 2.9 The Poincar´–Melnikov Method and Chaos e 101 of f . The estimates for the splitting have the form C(δ/µ2 ) exp(−π/ |µ|). Now consider x + µx + x2 + x3 = δf (t). ¨ (2.9.30) With δ = 0, there are equilibria at µ x = 0, −r, or − and x = 0, ˙ (2.9.31) r where √ 1+ 1 − 4µ r= , (2.9.32) 2 which is approximately 1 when µ ≈ 0. The phase portrait of (2.9.30) with δ = 0 and µ = − 1 is shown in Figure 2.9.2. As µ passes through 0, the 2 small lobe in Figure 2.9.2 undergoes the same bifurcation as in Figure 2.9.1, with the large lobe changing only slightly. . x x Figure 2.9.2. The phase portrait of x − 1 x + x2 + x3 = 0. ¨ 2 Again we rescale to give  δ τ  ¨ ξ − ξ + ξ 2 − µξ 3 = 2 f √ , µ < 0,    µ −µ  (2.9.33)   δ τ  ¨ 2 3 ξ + ξ + ξ + µξ = 2 f √ , µ > 0.   µ µ Notice that for δ = 0, the phase portrait is µ-dependent. The homoclinic orbit surrounding the small lobe for µ < 0 is given explicitly in terms of ξ ........................... 15 July 1998—18h02 ...........................

116. 102 2. Hamiltonian Systems on Linear Symplectic Spaces by 4eτ ξ(τ ) = , (2.9.34) 2 2 eτ + 3 − 2µ which is µ-dependent. An interesting technicality is that without the cubic term, we get µ-independent double poles at t = ±iπ + log 2 − log 3 in the complex τ -plane, while (2.9.34) has a pair of simple poles that splits these double poles to the pairs of simple poles at 2 √ τ = ±iπ + log ± i 2λ , (2.9.35) 3 where again λ = |µ|. (There is no particular signiﬁcance to the real part, such as log 2 − log 3 in the case of no cubic term; this can always be gotten rid of by a shift in the base point ξ(0).) If a quartic term x4 is added, these pairs of simple poles will split into quartets of branch points and so on. Thus, while the analysis of higher-order terms has this interesting µ-dependence, it seems that the basic exponential part of the estimates, namely π exp − , (2.9.36) |µ| remains intact. ........................... 15 July 1998—18h02 ...........................

117. This is page 103 Printer: Opaque this 3 An Introduction to Infinite-Dimensional Systems A common choice of configuration space for classical field theory is an infinite-dimensional vector space of functions or tensor fields on space or spacetime, the elements of which are called fields. Here we relate our treatment of infinite-dimensional Hamiltonian systems discussed in §2.1 to classical Lagrangian and Hamiltonian field theory and then give examples. Classical field theory is a large subject with many aspects not covered here; we treat only a few topics that are basic to subsequent developments; see Chapters 6 and 7 for additonal information and references. 3.1 Lagrange’s and Hamilton’s Equations for Field Theory As with finite-dimensional systems, one can begin with a Lagrangian and a variational principle, and then pass to the Hamiltonian via the Legendre transformation. At least formally, all the constructions we did in the finite- dimensional case go over to the infinite-dimensional one. For instance, suppose we choose our configuration space Q = F(R3 ) to be the space of fields ϕ on R3 . Our Lagrangian will be a function L(ϕ, ϕ) ˙ from Q × Q to R. The variational principle is b δ L(ϕ, ϕ) dt = 0, ˙ (3.1.1) a

118. 104 3. An Introduction to Infinite-Dimensional Systems which is equivalent to the Euler–Lagrange equations d δL δL = (3.1.2) dt δ ϕ ˙ δϕ in the usual way. Here, δL π= (3.1.3) δϕ ˙ is the conjugate momentum which we regard as a density on R3 , as in Chapter 2. The corresponding Hamiltonian is H(ϕ, π) = π ϕ − L(ϕ, ϕ) ˙ ˙ (3.1.4) in accordance with our general theory. We also know that the Hamiltonian should generate the canonical Hamilton equations. We verify this now. Proposition 3.1.1. Let Z = F(R3 ) × Den(R3 ), with Ω defined as in Example (b) of §2.2. Then the Hamiltonian vector field XH : Z → Z corresponding to a given energy function H : Z → R is given by δH δH XH = ,− . (3.1.5) δπ δϕ Hamilton’s equations on Z are ∂ϕ δH ∂π δH = , =− . (3.1.6) ∂t δπ ∂t δϕ Remarks. 1. The symbols F and Den stand for function spaces included in the space of all functions and densities, chosen appropriate to the functional analysis needs of the particular problem. In practice this often means, among other things, that appropriate conditions at infinity are imposed to permit integration by parts. 2. The equations of motion for a curve z(t) = (ϕ(t), π(t)) written in the form Ω(dz/dt, δz) = dH(z(t)) · δz for all δz ∈ Z with compact support, are called the weak form of the equations of motion. They can still be valid when there is not enough smoothness or decay at infinity to justify the literal equality dz/dt = XH (z); this situation can occur, for example, if one is considering shock waves. Proof of Proposition 3.1.1. To derive the partial functional derivatives, we use the natural pairing , : F(R3 ) × Den(R3 ) → R, where ϕ, π = ϕπ d3 x, (3.1.7) ........................... 15 July 1998—18h02 ...........................

119. 3.2 Examples: Hamilton’s Equations 105 where we write π = π d3 x ∈ Den. Recalling that δH/δϕ is a density, let δH δH X= ,− . δπ δϕ We need to verify that Ω(X(ϕ, π), (δϕ, δπ)) = dH(ϕ, π) · (δϕ, δπ). Indeed, δH δH Ω(X(ϕ, π), (δϕ, δπ)) = Ω ,− , (δϕ, δπ) δπ δϕ δH δH = (δπ) d3 x + δϕ d3 x δπ δϕ δH δH = , δπ + δϕ, δπ δϕ = Dπ H(ϕ, π) · δπ + Dϕ H(ϕ, π) · δϕ = dH(ϕ, π) · (δϕ, δπ). 3.2 Examples: Hamilton’s Equations (a) The Wave Equation. Consider Z = F(R3 ) × Den(R3 ) as above. Let ϕ denote the configuration variable, that is, the first component in the phase space F(R3 ) × Den(R3 ), and interpret ϕ as a measure of the displacement from equilibrium of a homogeneous elastic medium. Writing π = ρ dϕ/dt, where ρ is the mass density, the kinetic energy is insert argument saying 1 1 the change in T = [π ]2 d3 x. arc length is 2 ρ φ2 /2. x For small displacements ϕ, one assumes a linear restoring force such as the one given by the potential energy k 2 ϕ d3 x, 2 for an (elastic) constant k. Because we are considering a homogeneous medium, ρ and k are constants, so let us work in units in which they are unity. Nonlinear effects can be modeled in a naive way by introducing a nonlinear term, U (ϕ) into the potential. However, for an elastic medium one really should use constitutive relations based on the principles of continuum mechanics; see Marsden and Hughes [1983]. For the naive model, the Hamiltonian H : Z → R is the total energy 1 1 H(ϕ, π) = (π )2 + ϕ 2 + U (ϕ) d3 x. (3.2.1) 2 2 ........................... 15 July 1998—18h02 ...........................

120. 106 3. An Introduction to Infinite-Dimensional Systems Using the definition of the functional derivative, we find that δH δH 2 =π, = (− ϕ + U (ϕ))d3 x. (3.2.2) δπ δϕ Therefore, the equations of motion are ∂ϕ ∂π =π, = 2 ϕ − U (ϕ), (3.2.3) ∂t ∂t or, in second-order form, ∂2ϕ = 2 ϕ − U (ϕ). (3.2.4) ∂t2 Various choices of U correspond to various physical applications. When U = 0, we get the linear wave equation, with unit propagation velocity. Another choice, U (ϕ) = 1 m2 ϕ2 + λϕ4 , occurs in the quantum theory of 2 self-interacting mesons; the parameter m is related to the meson mass, and ϕ4 governs the nonlinear part of the interaction. When λ = 0, we get ∂2ϕ 2 ϕ− = m2 ϕ, (3.2.5) ∂t2 which is called the Klein-Gordon equation. Technical Aside. For the wave equation, one appropriate choice of function space is Z = H 1 (R3 ) × L2 (R3 ), where H 1 (R3 ) denotes the H 1 - Den functions on R3 , that is, functions which, along with their first derivatives, are square integrable, and L2 (R3 ) denotes the space of densities Den π = π d3 x, where the function π on R3 is square integrable. Note that the Hamiltonian vector field XH (ϕ, π) = (π , ( 2 ϕ − U (ϕ))d3 x) is defined only on the dense subspace H 2 (R3 ) × HDen (R3 ) of Z. This is a 1 common occurrence in the study of Hamiltonian partial differential equations; we return to this in §3.3. In the preceding example, Ω was given by the canonical form with the result that the equations of motion were in the standard form (3.1.5). In addition, the Hamiltonian function was given by the actual energy of the system under consideration. We now give examples in which these statements require reinterpretation but which nevertheless fall into the framework of the general theory developed so far. (b) The Schr¨dinger Equation. Let H be a complex Hilbert space, o for example, the space of complex-valued functions ψ on R3 with the inner product ψ1 , ψ2 = ψ1 (x)ψ 2 (x) d3 x, ........................... 15 July 1998—18h02 ...........................

121. 3.2 Examples: Hamilton’s Equations 107 where the overbar denotes complex conjugation. For a self-adjoint, complex- linear operator Hop : H → H, the Schr¨dinger equation is o ∂ψ i = Hop ψ, (3.2.6) ∂t where is Planck’s constant. Define −i A= Hop so that the Schr¨dinger equation becomes o ∂ψ = Aψ. (3.2.7) ∂t The symplectic form on H is given by Ω(ψ1 , ψ2 ) = −2 Im ψ1 , ψ2 . Self- adjointness of Hop is a condition stronger than symmetry and is essential for proving well-posedness of the initial-value problem for (3.2.6); for an exposition, see, for instance, Abraham, Marsden, and Ratiu [1988]. Histor- ically, it was Kato [1950] who established this for important problems such as the hydrogen atom. From §2.5, we know that since Hop is symmetric, A is Hamiltonian. The Hamiltonian is H(ψ) = iAψ, ψ = Hop ψ, ψ (3.2.8) which is the expectation value of Hop at ψ, defined by Hop (ψ) = Hop ψ, ψ . (c) The Korteweg-de Vries (KdV) Equation. Denote by Z the vector subspace F(R) consisting of those functions u with |u(x)| decreasing sufficiently fast as x → ±∞ so that the integrals we will write are defined and integration by parts is justified. As we shall see later, the Poisson brackets for the KdV equation are quite simple, and historically they were found first (see Gardner [1971] and Zakharov [1971, 1974]). To be consistent with our exposition, we begin with the somewhat more complicated symplectic structure. Pair Z with itself using the L2 inner product. Let the KdV symplectic structure Ω be defined by ∞ 1 Ω(u1 , u2 ) = [ˆ1 (x)u2 (x) − u2 (x)u1 (x)] dx , u ˆ (3.2.9) 2 −∞ where u denotes a primitive of u, that is, ˆ x u= ˆ u(y) dy. −∞ ........................... 15 July 1998—18h02 ...........................

122. 108 3. An Introduction to Inﬁnite-Dimensional Systems In §8.5 we shall see a way to construct this form. The form Ω is clearly skew-symmetric. Note that if u1 = ∂v/∂x for some v ∈ Z, then ∞ u2 (x)u1 (x) dx ˆ −∞ ∞ ∂ u1 (x) ˆ = u2 (x) ˆ dx −∞ ∂x ∞ ∞ = u1 (x)ˆ2 (x) ˆ u − u1 (x)u2 (x) dx ˆ −∞ −∞ ∞ ∞ ∞ ∂v(x) = dx u2 (x) dx − u1 (x)u2 (x) dx ˆ −∞ ∂x −∞ −∞ ∞ ∞ ∞ = v(x) u2 (x) dx − u1 (x)u2 (x) dx ˆ −∞ −∞ −∞ ∞ =− u1 (x)u2 (x) dx. ˆ −∞ Thus, if u1 (x) = ∂v(x)/∂x, then Ω can be written as ∞ ∞ Ω(u1 , u2 ) = u1 (x)u2 (x) dx = ˆ v(x)u2 (x) dx. (3.2.10) −∞ −∞ To prove weak nondegeneracy of Ω, we check that if v = 0, there is a w such that Ω(w, v) = 0. Indeed, if v = 0 and we let w = ∂v/∂x, then w = 0 because v(x) → 0 as |x| → ∞. Hence by (3.2.10), ∞ ∂v Ω(w, v) = Ω ,v = (v(x))2 dx = 0. ∂x −∞ Suppose that a Hamiltonian H : Z → R is given. We claim that the corresponding Hamiltonian vector ﬁeld XH is given by ∂ δH XH (u) = . (3.2.11) ∂x δu Indeed, by (3.2.10), ∞ δH Ω(XH (v), w) = (x)w(x) dx = dH(v) · w. −∞ δv It follows from (3.2.11) that the corresponding Hamilton equations are ∂ δH ut = , (3.2.12) ∂x δu where, in (3.2.12) and in the following, subscripts denote derivatives with respect to the subscripted variable. As a special case, consider the function ∞ 1 H1 (u) = − u3 dx. 6 −∞ ........................... 15 July 1998—18h02 ...........................

123. 3.2 Examples: Hamilton’s Equations 109 Then ∂ δH1 = −uux , ∂x δu and so (3.2.12) becomes the one-dimensional transport equation ut + uux = 0. (3.2.13) Next, let ∞ 1 2 H2 (u) = u − u3 dx; (3.2.14) −∞ 2 x then (3.2.12) becomes ut + 6uux + uxxx = 0. (3.2.15) This is the Korteweg-de Vries (KdV ) equation, describing shallow water waves. For a concise presentation of its famous complete set of integrals, see Abraham and Marsden [1978], §6.5, and for more information, see Newell [1985]. Traveling Waves. If we look for traveling wave solutions of (3.2.15), that is, u(x, t) = ϕ(x − ct), for a constant c > 0 and a positive function ϕ, we see that u satisﬁes the KdV equation if and only if ϕ satisﬁes cϕ − 6ϕϕ − ϕ = 0. (3.2.16) Integrating once gives cϕ − 3ϕ2 − ϕ = C, (3.2.17) where C is a constant. This equation is Hamiltonian in the canonical variables (ϕ, ϕ ) with Hamitonian function 1 c h(ϕ, ϕ ) = (ϕ )2 − ϕ2 + ϕ3 + Cϕ. (3.2.18) 2 2 From conservation of energy, h(ϕ, ϕ ) = D, it follows that ϕ = ± cϕ2 − 2ϕ3 − 2Cϕ + 2D, (3.2.19) or, writing s = x − ct, we get dϕ s=± . (3.2.20) cϕ2 − 2ϕ3 − 2Cϕ + 2D We seek solutions which, together with their derivatives vanish at ±∞. Then (3.2.17) and (3.2.19) give C = D = 0, so √ √ dϕ 1 c − 2ϕ − c s=± = ± √ log √ √ +K (3.2.21) cϕ2 − 2ϕ3 c c − 2ϕ + c ........................... 15 July 1998—18h02 ...........................

124. 110 3. An Introduction to Inﬁnite-Dimensional Systems for some constant K that will be determined below. For C = D = 0, the Hamiltonian (3.2.18) becomes 1 c h(ϕ, ϕ ) = (ϕ )2 − ϕ2 + ϕ3 (3.2.22) 2 2 and thus the two equilibria given by ∂h/∂ϕ = 0, ∂h/∂ϕ = 0, are (0, 0) and (c/3, 0). The matrix of the linearized Hamiltonian system at these equilibria is 0 1 ±c 0 which shows that (0, 0) is a saddle and (c/3, 0) is spectrally stable. The second variation criterion on the potential energy (see §1.10) − 2 ϕ2 + ϕ3 c at (c/3, 0) shows that this equilibrium is stable. Thus, if (ϕ(s), ϕ (s)) is a homoclinic orbit emanating and ending at (0, 0), the value of the Hamil- tonian function (3.2.22) on it is H(0, 0) = 0. From (3.2.22) it follows that (c/2, 0) is a point on this homoclinic orbit and thus (3.2.20) for C = D = 0 is its expression. Taking the initial condition of this orbit at s = 0 to be ϕ(0) = c/2, ϕ (0) = 0, (3.2.21) forces K = 0 and so √ √ c − 2ϕ − c √ √ √ = e± cs . c − 2ϕ + c Since ϕ ≥ 0 by hypothesis, the expression in the absolute value is negative and thus √ √ c − 2ϕ − c √ √ √ = −e± cs , c − 2ϕ + c whose solution is √ 2ce± cs c ϕ(s) = √ = √ . (1 + e± cs )2 2 cosh2 ( cs/2) This produces the soliton solution √ c c u(x, t) = sech2 (x − ct) . 2 2 (d) Sine-Gordon Equation. For functions u(x, t), where x and t are real variables, the sine-Gordon equation is utt = uxx + sin u. Equation (3.2.4) shows that it is Hamiltonian with π = ut dx, so π = ut , ∞ 1 2 1 2 H(u) = u + u + cos u dx, (3.2.23) −∞ 2 t 2 x and the canonical bracket structure, as in the wave equation. This equation also has a complete set of integrals; see again Newell [1985]. ........................... 15 July 1998—18h02 ...........................

125. 3.2 Examples: Hamilton’s Equations 111 (e) Abstract Wave Equation. Let H be a real Hilbert space and B : H → H a linear operator. On H × H, put the symplectic structure Ω given by (2.2.6). One can check that: 0 I (i) A = is Ω-skew if and only if B is a symmetric operator −B 0 on H; and (ii) if B is symmetric, then a Hamiltonian for A is 1 2 H(x, y) = ( y + Bx, x ). (3.2.24) 2 The equations of motion (2.4.10) give the abstract wave equation: x + Bx = 0. ¨ (f ) Linear Elastodynamics. On R3 consider the equations ρutt = div(c · u), that is, ∂ ∂uk ρui = tt cijkl l , (3.2.25) ∂xj ∂x where ρ is a positive function, and c is a fourth-order tensor field (the elasticity tensor ) on R3 with the symmetries cijkl = cklij = cjikl . On F(R3 ; R3 ) × F(R3 ; R3 ) (or more precisely on H 1 (R3 ; R3 ) × L2 (R3 ; R3 ) with suitable decay properties at infinity), define ˙ ˙ Ω((u, u), (v, v)) = ρ(v · u − u · v) d3 x. ˙ ˙ (3.2.26) R3 The form Ω is the canonical symplectic form (2.2.3) for fields u and their conjugate momenta π = ρu. ˙ On the space of functions u : R3 → R3 , consider the ρ-weighted L2 -inner product u, v ρ = ρu · v d3 x. (3.2.27) R3 Then the operator Bu = −(1/ρ) div(c · u) is symmetric with respect to ˙ this inner product and thus by Example (e) above, the operator A(u, u) = (u, (1/ρ) div(c · u)) is Ω-skew. ˙ ........................... 15 July 1998—18h02 ...........................

126. 112 3. An Introduction to Infinite-Dimensional Systems The equations (3.2.25) of linear elastodynamics are checked to be Hamil- tonian with respect to Ω given by (3.2.26), and with energy 1 2 1 ˙ H(u, u) = ˙ ρ u d3 x + cijkl eij ekl d3 x, (3.2.28) 2 2 where 1 ∂ui ∂uj eij = + . 2 ∂xj ∂xi Exercises 3.2-1. (a) Let ϕ : Rn+1 → R. Show directly that the sine-Gordon equation ∂2ϕ − 2 ϕ + sin ϕ = 0 ∂t2 are the Euler–Lagrange equations of a suitable Lagrangian. (b) Let ϕ : Rn+1 → C. Write the nonlinear Schr¨dinger equation o ∂ϕ 2 i + ϕ + βϕ|ϕ|2 = 0 ∂t as a Hamiltonian system. 3.2-2. Find a “soliton” solution for the sine-Gordon equation ∂2ϕ ∂2ϕ − + sin ϕ = 0 ∂t2 ∂x2 in one-spatial dimension. 3.2-3. Consider the complex nonlinear Schr¨dinger equation in one spa- o tial dimension ∂ϕ ∂ 2 ϕ i + + βϕ|ϕ|2 = 0, β = 0. ∂t ∂x2 (a) Show that the function ψ : R → C defining the traveling wave solution ϕ(x, t) = ψ(x − ct) for c > 0 satisfies a second-order complex differential equation equivalent to a Hamiltonian system in R4 relative to the non-canonical symplectic form whose matrix is given by   0 c 1 0  −c 0 0 1  Jc =   −1 0 0 0  .  0 −1 0 0 (See Exercise 2.4-1). ........................... 15 July 1998—18h02 ...........................

127. 3.3 Examples: Poisson Brackets and Conserved Quantities 113 (b) Analyze the equilibria of the resulting Hamiltonian system in R4 and determine their linear stability properties. (c) Let ψ(s) = eics/2 a(s) for a real function a(s) and determine a second- order equation for a(s). Show that the resulting equation is Hamilto- nian and has heteroclinic orbits for β < 0. Find them. (d) Find “soliton” solutions for the complex nonlinear Schr¨dinger equa- o tion. 3.3 Examples: Poisson Brackets and Conserved Quantities Before proceeding with infinite dimensional examples, it is first useful to recall some basic facts about angular momentum of particles in R3 . (The reader should supply a corresponding discussion for linear momentum.) Consider a particle moving in R3 under the influence of a potential V . Let the position coordinate be denoted q so that Newton’s second law reads m¨ = − V (q). q Let p = mq be the linear momentum and J = q × p be the angular ˙ momentum. Then d J = q × p + q × p = −q × ˙ ˙ V (q). dt If V is radially symmetric, it is a function of q alone: assume V (q) = f ( q 2 ), where f is a smooth function (exclude q = 0 if necessary). Then V (q) = 2f ( q 2 )q so that q × V (q) = 0. Thus, in this case, J is conserved. Alternatively, with 1 2 H(q, p) = p + V (q), 2m we can check directly that {H, Jl } = 0, where J = (J1 , J2 , J3 ) . Additional insight is gained by looking at the components of J more closely. For example, consider the scalar function F (q, p) = J(q, p) · ωk, ........................... 15 July 1998—18h02 ...........................

128. 114 3. An Introduction to Infinite-Dimensional Systems where ω is a constant, and k = (0, 0, 1). We find F (q, p) = ω(q 1 p2 − p1 q 2 ). The Hamiltonian vector field of F is ∂F ∂F ∂F ∂F ∂F ∂F XF (q, p) = , , ,− 1,− 2,− 3 ∂p1 ∂p2 ∂p3 ∂q ∂q ∂q = (−ωq 2 , ωq 1 , 0, −ωp2 , ωp1 , 0). We note that XF is just the vector field corresponding to the flow in the (q 1 , q 2 ) plane and the (p1 , p2 ) plane given by rotations about the origin with angular velocity ω. More generally, Jω := J · ω, where ω is a vector in R3 has Hamiltonian vector field whose flow consists of rotations about the axis ω. As we shall see later on in Chapters 11 and 12, this is the basis for understanding the link between conservation laws and symmetry more generally. Another identity is worth noting, namely, for two vectors ω 1 and ω 2 , {Jω1 , Jω2 } = Jω1 ×ω2 , which, as we shall see later, is an important link between the Poisson bracket structure and the structure of the Lie algebra of the rotation group. (a) The Schr¨dinger Bracket. In Example (b) of §3.2, we saw that if o Hop is a self-adjoint complex linear operator on a Hilbert space H, then A = Hop /i is Hamiltonian and the corresponding energy function HA is the expectation value Hop of Hop . Letting Hop and Kop be two such operators, and applying the Poisson bracket-commutator correspondence (2.7.10), or a direct calculation, we get { Hop , Kop } = [Hop , Kop ] . (3.3.1) In other words, the expectation value of the commutator is the Poisson bracket of the expectation values. Results like this leads one to statements like: “Commutators in quantum mechanics are not only analogous to Poisson brackets, they are Poisson brackets.” Even more striking are true statements like this “Don’t tell me that quantum mechanics is right and classical mechanics is wrong—after all quantum mechanics is a special case of classical mechanics.” Notice that if we take Kop ψ = ψ, the identity operator, the corresponding Hamiltonian function is p(ψ) = ψ 2 and from (3.3.1) we see that p is a conserved quantity for any choice of Hop , a fact that is central to the probabilistic interpretation of quantum mechanics. Later we shall see that p is the conserved quantity associated to the phase symmetry ψ → eiθ ψ. More generally, if F and G are two functions on H with δF/δψ = F , the gradient of F taken relative to the real inner product Re , on H, one ........................... 15 July 1998—18h02 ...........................

129. 3.3 Examples: Poisson Brackets and Conserved Quantities 115 finds that 1 XF = F (3.3.2) 2i and 1 {F, G} = − Im F, G . (3.3.3) 2 Notice that (3.3.2), (3.3.3), and Im z = − Re(iz) give 1 dF · XG = Re F, XG = Re F, −i G 2 1 = Re i F, G 2 1 = − Im F, G 2 = {F, G} as expected. (b) KdV Bracket. Using the definition of the bracket (2.7.1), the symplectic structure, and the Hamiltonian vector field formula from Exam- ple (c) of §3.2, one finds that ∞ δF ∂ δG {F, G} = dx (3.3.4) −∞ δu ∂x δu for functions F, G of u having functional derivatives that vanish at ±∞. (c) Linear and Angular Momentum for the Wave Equation. The wave equation on R3 discussed in Example (a) of §3.2 has the Hamil- tonian 1 1 H(ϕ, π) = (π )2 + ϕ 2 + U (ϕ) d3 x. (3.3.5) R3 2 2 Define the linear momentum in the x-direction by ∂ϕ 3 Px (ϕ, π) = π d x. (3.3.6) ∂x By (3.3.6), δPx /δπ = ∂ϕ/∂x, and δPx /δϕ = (−∂π /∂x) d3 x, so we get from (3.2.2) δPx δH δH δPx {H, Px }(ϕ, π) = − R3 δπ δϕ δπ δϕ ∂ϕ ∂π = (− 2 ϕ + U (ϕ)) + π d3 x R3 ∂x ∂x ∂ϕ ∂ 1 = − 2ϕ + U (ϕ) + (π )2 d3 x = 0 (3.3.7) R3 ∂x ∂x 2 ........................... 15 July 1998—18h02 ...........................

130. 116 3. An Introduction to Infinite-Dimensional Systems assuming the fields and U vanish appropriately at ∞. (The first term vanishes because it switches sign under integration by parts.) Thus, Px is conserved. The conservation of Px is connected with invariance of H under translations in the x-direction. Deeper insights into this connection are explored later. Of course, similar conservation laws hold in the y- and z-directions. Likewise, the angular momenta J = (Jx , Jy , Jz ), where, for example, ∂ ∂ Jz (ϕ) = π x −y ϕ d3 x (3.3.8) R3 ∂y ∂x are constants of the motion. This is proved in an analogous way. (For precise function spaces in which these operations can be justified, see Chernoff and Marsden [1974].) (d) Linear and Angular Momentum: the Schr¨dinger Equation. o Linear Momentum. In Example (b) of §3.2, assume that H is the space of complex-valued L2 -functions on R3 and that the self-adjoint linear operator Hop : H → H commutes with infinitesimal translations of the argument by a fixed vector ξ ∈ R3 , that is, Hop (Dψ(·) · ξ) = D(Hop ψ(·)) · ξ for any ψ whose derivative is in H. One checks, using (3.3.1) that i Pξ (ψ) = Dψ · ξ, ψ (3.3.9) Poisson commutes with Hop . If ξ is the unit vector along the x-axis, the corresponding conserved quantity is i ∂ψ Px (ψ) = ,ψ . ∂x Angular Momentum. Assume that Hop : H → H commutes with infinitesimal rotations by a fixed skew-symmetric 3 × 3 matrix ω , that is, ˆ Hop (Dψ(x) · ω x) = D((Hop ψ)(x)) · ω x ˆ ˆ (3.3.10) for every ψ whose derivative is in H, where, on the left-hand side, Hop is thought of as acting on the function x → Dψ(x) · ω x. Then the angular ˆ momentum function J(ˆ ) : x → iDψ(x) · ω (x)/ , ψ(x) ω ˆ (3.3.11) Poisson commutes with H so is a conserved quantity. If we choose ω = (0, 0, 1); that is,   0 −1 0 ω = 1 0 0 , ˆ 0 0 0 ........................... 15 July 1998—18h02 ...........................

131. 3.3 Examples: Poisson Brackets and Conserved Quantities 117 this corresponds to an infinitesimal rotation around the z-axis. Explicitly, the angular momentum around the xl -axis is given by i ∂ψ ∂ψ Jl (ψ) = xj − xk j ,ψ , ∂xk ∂x where (j, k, l) is a cyclic permutation of (1, 2, 3). (e) Linear and Angular Momentum for Linear Elastodynamics. Consider again the equations of linear elastodynamics; see Example (f) of §3.2. Observe that the Hamiltonian is invariant under translations if the elasticity tensor c is homogeneous (independent of (x, y, z)); the corresponding conserved linear momentum in the x-direction is ∂u 3 Px = ρu · ˙ d x. (3.3.12) R3 ∂x Likewise the Hamiltonian is invariant under rotations if c is isotropic; that is, invariant under rotations, which is equivalent to c having the form cijkl = µ(δ ik δ jl + δ il δ jk ) + λδ ij δ kl , where µ and λ are constants (see Marsden and Hughes [1983], §4.3, for the proof). The conserved angular momentum about the z-axis is ∂u ∂u J= ρu · x ˙ −y d3 x. R3 ∂y ∂x In Chapter 11, we will gain a deeper insight into the significance and construction of these conserved quantities. Some Technicalities for Infinite-Dimensional Systems. In general, unless the symplectic form on the Banach space Z is strong, the Hamil- tonian vector field XH is not defined on the whole of Z but only on a dense subspace. For example, in the case of the wave equation ∂ 2 ϕ/∂t2 = 2 ϕ − U (ϕ), a possible choice of phase space is H 1 (R3 ) × L2 (R3 ), but XH is defined only on the dense subspace H 2 (R3 ) × H 1 (R3 ). It can also happen that the Hamiltonian H is not even defined on the whole of Z. For example, if Hop = 2 + V for the Schr¨dinger equation on L2 (R3 ), then H o could have domain containing H 2 (R3 ) which coincides with the domain of the Hamiltonian vector field iHop . If V is singular, the domain need not be exactly H 2 (R3 ). As a quadratic form, H might be extendable to H 1 (R3 ). See Reed and Simon [1974, Volume II] or Kato [1984] for details. The problem of existence and even uniqueness of solutions can be quite delicate. For linear systems one often appeals to Stone’s theorem for the Schr¨dinger and wave equations, and to the Hille-Yosida theorem in the o ........................... 15 July 1998—18h02 ...........................

132. 118 3. An Introduction to Infinite-Dimensional Systems case of more general linear systems. We refer to Marsden and Hughes [1983], Chapter 6, for the theory and examples. In the case of nonlinear Hamilto- nian systems, the theorems of Segal [1962], Kato [1975], and Hughes, Kato, and Marsden [1977] are relevant. For infinite-dimensional nonlinear Hamiltonian systems technical differ- entiability conditions on its flow ϕt are needed to ensure that each ϕt is a symplectic map; see Chernoff and Marsden [1974], and especially Mars- den and Hughes [1983], Chapter 6. These technicalities are needed in many interesting examples. Exercises 3.3-1. Show that {Fi , Fj } = 0, i, j = 0, 1, 2, 3, where the Poisson bracket is the KdV bracket and where: ∞ F0 (u) = u dx −∞ ∞ 1 2 F1 (u) = u dx −∞ 2 ∞ 1 F2 (u) = −u3 + (ux )2 dx (the KdV Hamiltonian) −∞ 2 ∞ 5 4 1 F3 (u) = u − 5uu2 + x (uxx )2 dx. −∞ 2 2 ........................... 15 July 1998—18h02 ...........................

133. This is page 119 Printer: Opaque this 4 Interlude: Manifolds, Vector Fields, and Differential Forms In preparation for later chapters, it will be necessary for the reader to learn a little bit about manifold theory. We recall a few basic facts here, beginning with the finite-dimensional case. (See Abraham, Marsden, and Ratiu [1988] for a full account.) The reader need not master all of this material now, but it suffices to read through it for general sense and come back to it repeatedly as our development of mechanics proceeds. 4.1 Manifolds Coordinate Charts. Given a set M , a chart on M is asubset U of M together with a bijective map ϕ : U → ϕ(U ) ⊂ Rn . Usually we denote by (x1 , . . . xn ) = ϕ(m), the coordinates of a point m ∈ U ⊂ M . Two charts (U, ϕ) and (U , ϕ ) such that U ∩ U = 0 are called compatible, if ϕ(U ∩ U )) and ϕ(U ∩ U )) are open subsets of Rn and the maps ϕ ◦ ϕ−1 |ϕ(U ∩ U ) : ϕ(U ∩ U ) −→ ϕ (U ∩ U ), ϕ ◦ (ϕ )−1 |ϕ (U ∩ U ) : ϕ (U ∩ U ) −→ ϕ (U ∩ U ) are C ∞ . We call M a differentiable manifold if the following hold: M1. It is covered by a collection of charts, that is, every point is represented in at least one chart.

134. 120 4. Interlude: Manifolds, Vector Fields, and Differential Forms M2. M has an atlas; that is, M can be written as a union of compatible charts. xn ϕ(U) M ϕ x1 U xn ϕ (U ) U V ϕ x1 Figure 4.1.1. Overlapping charts on a manifold. Two atlases are called equivalent if their union is also an atlas. One often rephrases the definition by saying that a differentiable structure on a manifold is an equivalence class of atlases. A neighborhood of a point m in a manifold M is the image under the inverse of a chart map ϕ−1 : V → M of a neighborhood V of the representation of m ∈ M in a chart U . Neighborhoods define open sets and one checks that the open sets in M define a topology. Usually we assume without explicit mention that the topology is Hausdorff : two different points m, m in M have nonintersecting neighborhoods. A differentiable manifold M is called an n-manifold if every chart has domain in an n-dimensional vector space. Tangent Vectors. Two curves t → c1 (t) and t → c2 (t) in an n-manifold M are called equivalent at m if c1 (0) = c2 (0) = m and (ϕ ◦ c1 ) (0) = (ϕ ◦ c2 ) (0) in some chart ϕ. It is easy to check that this definition is chart independent. A tangent vector v to a manifold M at a point m ∈ M is an equivalence class of curves at m. One proves that the set of tangent vectors to M at m forms a vector space. It is denoted T M and is called the tangent space to M at m ∈ M . Given a curve c(t), we denote by c (s) the tangent vector at c(s) defined by the equivalence class of t → c(s + t) at t = 0. ........................... 15 July 1998—18h02 ...........................

135. 4.1 Manifolds 121 Let U be a chart of an atlas for the manifold M with coordinates (x1 , . . . , xn ). The components of the tangent vector v to the curve t → (ϕ ◦ c)(t) are the numbers v 1 , . . . , v n defined by d vi = (ϕ ◦ c)i , dt t=0 where i = 1, . . . , n. The tangent bundle of M , denoted by T M , is the differentiable manifold whose underlying set is the disjoint union of the tangent spaces to M at the points m ∈ M , that is, TM = Tm M. m∈M Thus, a point of T M is a vector v that is tangent to M at some point m ∈ M . To define the differentiable structure on T M , we need to specify how to construct local coordinates on T M . To do this, let x1 , . . . , xn be local coordinates on M and let v 1 , . . . , v n be components of a tangent vector in this coordinate system. Then the 2n numbers x1 , . . . , xn , v 1 , . . . , v n give a local coordinate system on T M . Notice that dim T M = 2 dim M . The natural projection is the map τM : T M → M that takes a tangent vector v to the point m ∈ M at which the vector v is attached (that is, −1 v ∈ Tm M ). The inverse image τM (m) of a point m ∈ M under the natural projection τM is the tangent space Tm M . This space is called the fiber of the tangent bundle over the point m ∈ M . Differentiable Maps. Let f : M → N be a map of a manifold M to a manifold N . We call f differentiable (or C k ) if in local coordinates on M and N it is given by differentiable (or C k ) functions. The derivative of a differentiable map f : M → N at a point m ∈ M is defined to be the linear map Tm f : Tm M → Tf (m) N constructed in the following way. For v ∈ Tm M , choose a curve c : ]− , [ → M with c(0) = m, and velocity vector dc/dt |t=0 = v . Then Tm f · v is the velocity vector at t = 0 of the curve f ◦ c : R → N , that is, d Tm f · v = f (c(t)) . dt t=0 The vector Tm f · v does not depend on the curve c but only on the vector v. If M and N are manifolds and f : M → N is of class C r+1 , then T f : T M → T N is a mapping of class C r . Note that dc = T0 c · 1. dt t=0 ........................... 15 July 1998—18h02 ...........................

136. 122 4. Interlude: Manifolds, Vector Fields, and Differential Forms A differentiable (or of class C r ) map f : M → N is called a diffeomorphism if it is bijective and its inverse is also differentiable (or of class C r ). If f : M → N and g : N → P are differentiable maps (or maps of class C r ), then g ◦ f : M → P is differentiable (or of class C r ) and the chain rule holds T (g ◦ f ) = T g ◦ T f. If Tm f : Tm M → Tf (m) N is an isomorphism, the Inverse Function Theorem states that f is a local diffeomorphism around m ∈ M , that is, there are open neighborhoods U of m in M and V of f (m) in N such that f |V : U → V is a diffeomorphism. Submanifolds and Submersions. A submanifold of M is a subset S ⊂ M with the property that for each s ∈ S there is a chart (U, ϕ) in M with the submanifold property , namely, SM. ϕ : U → Rk × Rn−k and ϕ(U ∩ S) = ϕ(U ) ∩ (Rk × {0}). The number k is called the dimension of the submanifold S. This latter notion is in agreement with the definition of dimension for a general manifold, since S is a manifold in its own right all of whose charts are of the form (U ∩ S, ϕ|U ∩ S) for all charts (U, ϕ) of M having the submanifold property. Note that any open subset of M is a submanifold and that a submanifold is necessarily locally closed , that is, every point s ∈ S admits an open neighborhood U of s in M such that U ∩ S is closed in U . It turns out that there are convenient ways to construct submanifolds using smooth mappings. If f : M → N is a smooth map, a point m ∈ M is a regular point if Tm f is surjective; otherwise m is a critical point of f . If C ⊂ M is the set of critical points of f , then f (C) ⊂ N is the set of critical values of f and N f (C) is the set of regular values of f . Sard’s Theorem states that if f : M → N is a C r -map, r ≥ 1, and if M has the property that every open covering has a countable subcovering, then if r > max(0, dim M −dim N ), the set of regular values of f is residual and hence dense in N . The Submersion Theorem statest that if f : M → N is a smooth map and n is a regular value of f , then f −1 (n) is a smooth submanifold of M of dimension dim M − dim N and Tm f −1 (n) = ker Tm f. The Local Onto Theorem states that Tm f : Tm M → Tf (m) N is surjective, if and only if there are charts (U, ϕ) at m in M and (V, ψ) at f (m) in N such that ϕ(U ) = U × V , ψ(V ) = V , ϕ(m) = (0, 0), ϕ(f (m)) = 0, (ψ ◦ f ◦ ϕ−1 )(x, y) = x. ........................... 15 July 1998—18h02 ...........................

137. 4.1 Manifolds 123 In particular, f |U : U → V is onto. If Tm f is onto for every m ∈ M , f is called a submersion. Submersions are open mappings. Immersions and Embeddings. A C r map of f : M → N is called an immersion if Tm f is injective for every m ∈ M . The Local 1-to-1 Theorem states that Tm f is injective, if and only if there are charts (U, ϕ) at m ∈ M , (V, ψ) at f (m) in N such that ϕ:U →U ψ :V →U ×V , ϕ(m) = 0, ψ(f (m)) = (0, 0), (ψ ◦ f ◦ ϕ−1 )(x) = (x, 0). In particular, f |U : U → V is injective. The Immersion Theorem states that Tm f is injective, if and only if there is a neighborhood U of m in M such that f (U ) is a submanifold of N and f |U : U → f (U ) is a difeomor- phism. It should be noted that this theorem does not say that f (M ) is a submanifold of N . For example, f may not be injective and f (M ) may thus have self-intersections. But even if f is an injective immersion, the image f (M ) may not be a submanifold of N . For example, the map whose graph is shown in Figure 4.1.2. is an injective immersion but the topology in- y f R2 a b x r = cos 2θ Figure 4.1.2. An injective immersion. duced from R2 onto its image does not coincide with the usual topology of the open interval: any neighborhood of the origin in the relative topology consists of the union of an open interval with two open rays ] − ∞, a[, ]b, ∞[. Thus the image of f is not a submanifold fo R2 , but an injectively immersed submanifold . An immersion f : M → N that is a homeomorphism onto f (M ) with the relative topology induced from N is called an embedding . In this case f (M ) is a submanifold of N and f : M → f (M ) is a diﬀeomorphism. Another example of an injective immersion that is not an embedding is the linear ﬂow on the torus T2 = R2 /Z2 with irrational slope: f (t) = ........................... 15 July 1998—18h02 ...........................

138. 124 4. Interlude: Manifolds, Vector Fields, and Differential Forms (t, αt) ( mod Z2 ). However, there is a fundamental difference between this injective immersion and the one described above: in some sense, the second example is better behaved; it has some “uniformity” about its lack of being an embedding. An injective immersion f : M → N is called regular if the following preoperty holds: if g : L → M is any map of the manifold L into M then g is C r if and only if f ◦ g : L → N is C r . It is easy to see that all embeddings satisfy this property but that the previous example also satisfies it, without being an embedding, and that the “figure eight” example (see Figure 4.1.2) does not satisfy it. Varadsajan [1984] calls such maps quasi-regular embeddings. They appear below in the Frobenius Theorem and in the study of Lie subgroups. Vector Fields and Flows. A vector field X on a manifold M is a map X : M → T M that assigns a vector X(m) at the point m ∈ M ; that is, τM ◦ X = identity. The real vector space of vector fields on M is denoted by X(M ). An integral curve of X with initial condition m0 at t = 0 is a (differentiable) map c : ]a, b[ → M such that ]a, b[ is an open interval containing 0, c(0) = m0 , and c (t) = X(c(t)) for all t ∈ ]a, b[. In formal presentations we usually suppress the domain of definition, even though this is technically important. The flow of X is the collection of maps ϕt : M → M such that t → ϕt (m) is the integral curve of X with initial condition m. Existence and uniqueness theorems from ordinary differential equations guarantee ϕ is smooth in m and t (where defined) if X is. From uniqueness, we get the flow property ϕt+s = ϕt ◦ ϕs along with the initial conditions ϕ0 = identity. The flow property generalizes the situation where M = V is a linear space, X(m) = Am for a (bounded) linear operator A, and where ϕt (m) = etA m to the nonlinear case. A time dependent vector field is a map X : M × R → T M such that X(m, t) ∈ Tm M for each m ∈ M and t ∈ R. An integral curve of X is a curve c(t) in M such that c (t) = X(c(t), t). In this case, the flow is the collection of maps ϕt,s : M → M such that t → ϕt,s (m) is the integral curve c(t) with initial condition c(s) = m at t = s. Again, the existence and uniqueness theorem from ODE theory ........................... 15 July 1998—18h02 ...........................

139. 4.1 Manifolds 125 applies and, in particular, uniqueness gives the time dependent flow property : ϕt,s ◦ ϕs,r = ϕt,r . If X happens to be time independent, the two notions of flows are related by ϕt,s = ϕt−s . Differentials and Covectors. If f : M → R is a smooth function, we can differentiate it at any point m ∈ M to obtain a map Tm f : Tm M → Tf (m) R. Identifying the tangent space of R at any point with itself (a process we usually do in any vector space), we get a linear map ∗ df (m) : Tm M → R. That is, df (m) ∈ Tm M , the dual of the vector space Tm M . In coordinates, the directional derivatives, defined by df (m) · v, for v ∈ Tm M , are given by n ∂(f ◦ ϕ−1 ) i df (m) · v = v. i=1 ∂xi where ϕ is a chart at m. We will employ the summation convention and drop the summation sign when there are repeated indices. We also call df the differential of f . One can show that specifying the directional derivatives completely determines a vector and so we can identify a basis of Tm M using the operators ∂/∂xi . We write ∂ ∂ {e1 , . . . , en } = ,... , n ∂x1 ∂x for this basis so that v = v i ∂/∂xi . ∗ If we replace each vector space Tm M with its dual Tm M , we obtain a new 2n-manifold called the cotangent bundle and denoted T ∗ M . The dual basis to ∂/∂xi is denoted dxi . Thus, relative to a choice of local coordinates we get the basic formula ∂f i df (x) = dx ∂xi for any smooth function f : M → R. Exercises 4.1-1. Show that the two-sphere S 2 ⊂ R3 is a 2-manifold. 4.1-2. If ϕt : S 2 → S 2 rotates points on S 2 about a fixed axis through an angle t, show that ϕt is the flow of a certain vector field on S 2 . 4.1-3. Let f : S 2 → R be defined by f (x, y, z) = z. Compute df relative to spherical coordinates (θ, ϕ). ........................... 15 July 1998—18h02 ...........................

140. 126 4. Interlude: Manifolds, Vector Fields, and Differential Forms 4.2 Differential Forms We next review some of the basic definitions, properties, and operations on differential forms, without proofs (see Abraham, Marsden, and Ratiu [1988] and references therein). The main idea of differential forms is to provide a generalization of the basic operations of vector calculus, div, grad, and curl, and the integral theorems of Green, Gauss, and Stokes to manifolds of arbitrary dimension. Basic Definitions. A 2-form Ω on a manifold M is a function Ω(m) : Tm M × Tm M → R that assigns to each point m ∈ M a skew-symmetric bilinear form on the tangent space Tm M to M at m. More generally, a k- form α (sometimes called a differential form of degree k) on a manifold M is a function α(m) : Tm M × . . . × Tm M (there are k factors) → R that assigns to each point m ∈ M a skew-symmetric k-multilinear map on the tangent space Tm M to M at m. Without the skew-symmetry assumption, α would be called a (0, k)-tensor . A map α : V × . . . × V (there are k factors) → R is multilinear when it is linear in each of its factors, that is, α(v1 , . . . , avj + bvj , . . . , vk ) = aα(v1 , . . . , vj , . . . , vk ) + bα(v1 , . . . , vj , . . . , vk ) for all j with 1 ≤ j ≤ k. A k-multilinear map α : V × . . . × V → R is skew (or alternating ) when it changes sign whenever two of its arguments are interchanged, that is, for all v1 , . . . , vk ∈ V , α(v1 , . . . , vi , . . . , vj , . . . , vk ) = −α(v1 , . . . , vj , . . . , vi , . . . , vk ). Let x1 , . . . , xn denote coordinates on M , let {e1 , . . . , en } = {∂/∂x1 , . . . , ∂/∂xn } be the corresponding basis for Tm M , and let {e1 , . . . , en } = {dx1 , . . . , dxn } ∗ be the dual basis for Tm M . Then at each m ∈ M , we can write a 2-form as ∂ ∂ Ωm (v, w) = Ωij (m)v i wj , where Ωij (m) = Ωm , i ∂xj , ∂x and, more generally, a k-form can be written i i αm (v1 , . . . , vk ) = αi1 ...ik (m)v11 . . . vkk , where there is a sum on i1 , . . . , ik and where ∂ ∂ αi1 ...ik (m) = αm ,... , i , ∂xi1 ∂x k j and where vi = vi ∂/∂xj , with a sum on j. ........................... 15 July 1998—18h02 ...........................

141. 4.2 Differential Forms 127 Tensor and Wedge Products. If α is a (0, k)-tensor on a manifold M , and β is a (0, l)-tensor, their tensor product α ⊗ β is the (0, k + l)-tensor on M defined by (α ⊗ β)m (v1 , . . . , vk+l ) = αm (v1 , . . . , vk )βm (vk+1 , . . . , vk+l ) (4.2.1) at each point m ∈ M . If t is a (0, p)-tensor, define the alternation operator A acting on t by 1 A(t)(v1 , . . . , vp ) = sgn(π)t(vπ(1) , . . . , vπ(p) ), (4.2.2) p! π∈Sp where sgn(π) is the sign of the permutation π: +1 if π is even, sgn(π) = (4.2.3) −1 if π is odd, and Sp is the group of all permutations of the set {1, 2, . . . , p}. The operator A therefore skew-symmetrizes p-multilinear maps. If α is a k-form and β is an l-form on M , their wedge product α ∧ β is the (k + l)-form on M defined by1 (k + l)! α∧β = A(α ⊗ β). (4.2.4) k! l! For example, if α and β are one-forms, (α ∧ β)(v1 , v2 ) = α(v1 )β(v2 ) − α(v2 )β(v1 ) while if α is a 2-form and β is a 1-form, (α ∧ β)(v1 , v2 , v3 ) = α(v1 , v2 )β(v3 ) + α(v3 , v1 )β(v2 ) + α(v2 , v3 )β(v1 ). We state the following without proof: Proposition 4.2.1. The wedge product has the following properties: (i) α ∧ β is associative: α ∧ (β ∧ γ) = (α ∧ β) ∧ γ. (ii) α ∧ β is bilinear in α, β : (aα1 + bα2 ) ∧ β = a(α1 ∧ β) + b(α2 ∧ β), α ∧ (cβ1 + dβ2 ) = c(α ∧ β1 ) + d(α ∧ β2 ). 1 The numerical factor in (4.2.4) agrees with the convention of Abraham and Marsden [1978], Abraham, Marsden, and Ratiu [1988], and Spivak [1976], but not that of Arnold [1989], Guillemin and Pollack [1974], or Kobayashi and Nomizu [1963]; it is the Bourbaki [1971] convention. ........................... 15 July 1998—18h02 ...........................

142. 128 4. Interlude: Manifolds, Vector Fields, and Differential Forms (iii) α∧β is anticommutative: α∧β = (−1)kl β ∧α, where α is a k-form and β is an l-form. In terms of the dual basis dxi , any k-form can be written locally as α = αi1 ...ik dxi1 ∧ · · · ∧ dxik where the sum is over all ij satisfying i1 < · · · < ik . Pull Back and Push Forward. Let ϕ : M → N be a C ∞ map from the manifold M to the manifold N and α be a k-form on N . Define the pull back ϕ∗ α of α by ϕ to be the k-form on M given by (ϕ∗ α)m (v1 , . . . , vk ) = αϕ(m) (Tm ϕ · v1 , . . . , Tm ϕ · vk ). (4.2.5) If ϕ is a diffeomorphism, the push forward ϕ∗ is defined by ϕ∗ = (ϕ−1 )∗ . Here is another basic property. Proposition 4.2.2. The pull back of a wedge product is the wedge product of the pull backs: ϕ∗ (α ∧ β) = ϕ∗ α ∧ ϕ∗ β. (4.2.6) Interior Products and Exterior Derivatives. Let α be a k-form on a manifold M and X a vector field. The interior product iX α (sometimes called the contraction of X and α, and written X α) is defined by (iX α)m (v2 , . . . , vk ) = αm (X(m), v2 , . . . , vk ). (4.2.7) Proposition 4.2.3. Let α be a k-form and β an l-form on a manifold M . Then iX (α ∧ β) = (iX α) ∧ β + (−1)k α ∧ (iX β). (4.2.8) In the ‘hook’ notation, this reads X (α ∧ β) = (X α) ∧ β + (−1)k α ∧ (X β). The exterior derivative dα of a k-form α on a manifold M is the (k + 1)- form on M determined by the following proposition: Proposition 4.2.4. There is a unique mapping d from k-forms on M to (k + 1)-forms on M such that: (i) If α is a 0-form (k = 0), that is, α = f ∈ F(M ), then df is the one-form which is the differential of f . (ii) dα is linear in α, that is, for all real numbers c1 and c2 , d(c1 α1 + c2 α2 ) = c1 dα1 + c2 dα2 . ........................... 15 July 1998—18h02 ...........................

143. 4.2 Differential Forms 129 (iii) dα satisfies the product rule, that is, d(α ∧ β) = dα ∧ β + (−1)k α ∧ dβ, where α is a k-form and, β is an l-form. (iv) d2 = 0, that is, d(dα) = 0 for any k-form α. (v) d is a local operator , that is, dα(m) only depends on α restricted to any open neighborhood of m; in fact, if U is open in M , then d(α|U ) = (dα)|U. If α is a k-form given in coordinates by α = αi1 ...ik dxi1 ∧ · · · ∧ dxik (sum on i1 < · · · < ik ), then the coordinate expression for the exterior derivative is ∂αi1 ...ik j dα = dx ∧ dxi1 ∧ · · · ∧ dxik ∂xj (sum on all j and i1 < · · · < ik ) (4.2.9) Formula (4.2.9) can be taken as the definition of the exterior derivative, provided one shows that (4.2.9) has the above-described properties and, correspondingly, is independent of the choice of coordinates. Next is a useful proposition that, in essence, rests on the chain rule: Proposition 4.2.5. Exterior differentiation commutes with pull back, that is, d(ϕ∗ α) = ϕ∗ (dα), (4.2.10) where α is a k-form on a manifold N and ϕ : M → N is a smooth map between manifolds. A k-form α is called closed if dα = 0 and exact if there is a (k − 1)- form β such that α = dβ. By Proposition 4.2.4iv every exact form is closed. Exercise 4.4-2 gives an example of a closed nonexact one-form. Proposition 4.2.6 (Poincar´ Lemma). A closed form is locally exact, e that is, if dα = 0 there is a neighborhood about each point on which α = dβ. See Exercise 4.2-5 for the proof. The definition and properties of vector-valued forms are direct extensions of these for usual forms on vector spaces and manifolds. One can think of a vector-valued form as an array of usual forms (see Abraham, Marsden, and Ratiu [1988]). ........................... 15 July 1998—18h02 ...........................

144. 130 4. Interlude: Manifolds, Vector Fields, and Differential Forms Vector Calculus. The table below entitled “Vector calculus and differential forms” summarizes how forms are related to the usual operations of vector calculus. We now elaborate on a few items in this table. In item 4, note that ∂f ∂f ∂f df = dx + dy + dz = (gradf ) = ( f ) ∂x ∂y ∂z which is equivalent to f = (df ) . The Hodge star operator on R3 maps k-forms to (3 − k)-forms and is uniquely determined by linearity and the properties in item 2. (This operator can be defined on general Riemannian manifolds; see Abraham, Mars- den, and Ratiu [1988].) In item 5, if we let F = F1 e1 +F2 e2 +F3 e3 , so F = F1 dx+F2 dy +F3 dz, then, d(F ) = dF1 ∧ dx + F1 d(dx) + dF2 ∧ dy + F2 d(dy) + dF3 ∧ dz + F3 d(dz) ∂F1 ∂F1 ∂F1 = dx + dy + dz ∧ dx ∂x ∂y ∂z ∂F2 ∂F2 ∂F2 + dx + dy + dz ∧ dy ∂x ∂y ∂z ∂F3 ∂F3 ∂F3 + dx + dy + dz ∧ dz ∂x ∂y ∂z ∂F1 ∂F1 ∂F2 ∂F2 =− dx ∧ dy + dz ∧ dx + dx ∧ dy − dy ∧ dz ∂y ∂z ∂x ∂z ∂F3 ∂F3 − dz ∧ dx + dy ∧ dz ∂x ∂y ∂F2 ∂F1 ∂F1 ∂F3 = − dx ∧ dy + − dz ∧ dx ∂x ∂y ∂z ∂x ∂F3 ∂F2 + − dy ∧ dz. ∂y ∂z Hence, using item 2, ∂F2 ∂F1 ∂F1 ∂F3 ∂F3 ∂F2 ∗(d(F )) = − dz + − dy + − dx, ∂x ∂y ∂z ∂x ∂y ∂z ∂F3 ∂F2 ∂F1 ∂F3 ∂F2 ∂F1 (∗(d(F ))) = − e1 + − e2 + − e3 ∂y ∂z ∂z ∂x ∂x ∂y = curl F = × F. With reference to item 6, let F = F1 e1 + F2 e2 + F3 e3 , so F = F1 dx + F2 dy + F3 dz. ........................... 15 July 1998—18h02 ...........................

145. 4.2 Diﬀerential Forms 131 Thus ∗(F ) = F1 dy ∧ dz + F2 (−dx ∧ dz) + F3 dx ∧ dy, and so d(∗(F )) = dF1 ∧ dy ∧ dz − dF2 ∧ dx ∧ dz + dF3 ∧ dx ∧ dy ∂F1 ∂F1 ∂F1 = dx + dy + dz ∧ dy ∧ dz ∂x ∂y ∂z ∂F2 ∂F2 ∂F2 − dx + dy + dz ∧ dx ∧ dz ∂x ∂y ∂z ∂F3 ∂F3 ∂F3 + dx + dy + dz ∧ dx ∧ dy ∂x ∂y ∂z ∂F1 ∂F2 ∂F3 = dx ∧ dy ∧ dz + dx ∧ dy ∧ dz + dx ∧ dy ∧ dz ∂x ∂y ∂z ∂F1 ∂F2 ∂F3 = + + dx ∧ dy ∧ dz = (div F ) dx ∧ dy ∧ dz. ∂x ∂y ∂z Therefore, ∗(d(∗(F ))) = div F = · F. Vector Calculus and Diﬀerential Forms 1. Sharp and Flat (Using standard coordinates in R3 ) (a) v = v 1 dx + v 2 dy + v 3 dz = one-form corresponding to the vector v = v 1 e1 + v 2 e2 + v 3 e3 . (b) α = α1 e1 + α2 e2 + α3 e3 = vector corresponding to the one-form α = α1 dx + α2 dy + α3 dz. 2. Hodge Star Operator (a) ∗1 = dx ∧ dy ∧ dz. (b) ∗dx = dy ∧ dz, ∗dy = −dx ∧ dz, ∗dz = dx ∧ dy, ∗(dy ∧ dz) = dx, ∗(dx ∧ dz) = −dy, ∗(dx ∧ dy) = dz. (c) ∗(dx ∧ dy ∧ dz) = 1. 3. Cross Product and Dot Product (a) v × w = [∗(v ∧ w )] . (b) (v · w)dx ∧ dy ∧ dz = v ∧ ∗(w ). 4. Gradient f = gradf = (df ) . 5. Curl × F = curl F = [∗(dF )] . 6. Divergence · F = div F = ∗d(∗F ). ........................... 15 July 1998—18h02 ...........................

146. 132 4. Interlude: Manifolds, Vector Fields, and Diﬀerential Forms Exercises 4.2-1. Let ϕ : R3 → R2 be given by ϕ(x, y, z) = (x + z, xy). For α = ev du + u dv ∈ Ω1 (R2 ) and β = u du ∧ dv, compute α ∧ β, ϕ∗ α, ϕ∗ β, and ϕ∗ α ∧ ϕ∗ β. 4.2-2. Given α = y 2 dx ∧ dz + sin(xy) dx ∧ dy + ex dy ∧ dz ∈ Ω2 (R3 ) and X = 3∂/∂x + cos z∂/∂y − x2 ∂/∂z ∈ X(R3 ), compute dα and iX α. 4.2-3. (a) Denote by ∧k (Rn ) the vector space of all skew-symmetric k-linear There’s no maps on Rn . Prove that this space has dimension n!/k! (n − k)! by deﬁnition for showing that a basis is given by {ei1 ∧ · · · ∧ eik | i1 < . . . < ik }, where “big” wedge. {e1 , . . . , en } is a basis of Rn and {e1 , . . . , en } is its dual basis, that How big would is, ei (ej ) = δj .i the wedge be? (b) If µ ∈ ∧n (Rn ) is nonzero, prove that the map v ∈ Rn → iv µ ∈ ∧n−1 (Rn ) is an isomorphism. (c) If M is a smooth n-manifold and µ ∈ Ωn (M ) is nowhere vanishing (in which case it is called a volume form), show that the map X ∈ X(M ) → iX µ ∈ Ωn−1 (M ) is a module isomorphism over F(M ). 4.2-4. Let α = αi dxi be a closed one-form in a ball around the origin in Rn . Show that α = df for 1 f (x1 , . . . , xn ) = αj (tx1 , . . . , txn )xj dt. 0 4.2-5. (a) Let U be an open ball around the origin in Rn and α ∈ Ωk (U ) a closed form. Verify that α = dβ, where β(x1 , . . . , xn ) 1 = tk−1 αji1 ...ik−1 (tx1 , . . . , txn )xj dt dxi1 ∧ . . . ∧ dxik−1 , 0 ........................... 15 July 1998—18h02 ...........................

147. 4.3 The Lie Derivative 133 and where the sum is over i1 < · · · < ik−1 . Here, α = αj1 ...jk dxj1 ∧ . . . ∧ dxjk , where j1 < · · · < jk and where α is extended to be skew-symmetric in its lower indices. (b) Deduce the Poincar´ lemma from (a). e 4.2-6. (Construction of a homotopy operator for a retraction.) Let M be a smooth manifold and N ⊂ M a smooth submanifold. A family of smooth maps rt : M → M, t ∈ [0, 1], is called a retraction of M onto N , if rt |N = identity on N for all t ∈ [0, 1], r1 = identity on M , rt is a diffeomorphism of M with rt (M ) for every t = 0, and r0 (M ) = N . Let Xt be the time dependent vector field generated by rt , t = 0. Show that the operator H : Ωk (M ) → Ωk−1 (M ) defined by 1 ∗ H= (rt iXt α) dt 0 satisfies ∗ α − (r0 α) = dHα + Hdα. (a) Deduce the relative Poincar´ lemma from this formula: if α ∈ e Ωk (M ) is closed and α|N = 0, then there is a neigborhood U of N such that α|U = dβ, for some β ∈ Ωk−1 (U ) and β|N = 0. (Hint: Use the existence of a tubular neigborhood of N in M .). (b) Deduce the global Poincar´ Lemma for contractible manifolds: If e M is contractible, that is, there is a retraction of M to a point, and if α ∈ Ωk (M ) is closed, then α is exact. 4.3 The Lie Derivative Lie Derivative Theorem. The dynamic definition of the Lie derivative is as follows. Let α be a k-form and let X be a vector field with flow ϕt . The Lie derivative of α along X is given by 1 d ∗ £X α = lim [(ϕ∗ α) − α] = t ϕ α . (4.3.1) t→0 t dt t t=0 This definition together with properties of pull-backs yields the following. Theorem 4.3.1 (Lie Derivative Theorem). d ∗ ϕ α = ϕ∗ £X α. (4.3.2) dt t t ........................... 15 July 1998—18h02 ...........................

148. 134 4. Interlude: Manifolds, Vector Fields, and Differential Forms This formula holds also for time-dependent vector fields in the sense that d ∗ ϕ α = ϕ∗ £ X α dt t,s t,s and in £X α, the vector field is evaluated at time t. If f is a real-valued function on a manifold M and X is a vector field on M , the Lie derivative of f along X is the directional derivative £X f = X[f ] := df · X. (4.3.3) If M is finite-dimensional, ∂f £X f = X i . (4.3.4) ∂xi For this reason one often writes ∂ X = Xi . ∂xi If Y is a vector field on a manifold N and ϕ : M → N is a diffeomorphism, the pull back ϕ∗ Y is a vector field on M defined by (ϕ∗ Y )(m) = Tm ϕ−1 ◦ Y ◦ ϕ(m). (4.3.5) Two vector fields X on M and Y on N are said to be ϕ-related if T ϕ ◦ X = Y ◦ ϕ. (4.3.6) Clearly, if ϕ : M → N is a diffeomorphism and Y is a vector field on N , ϕ∗ Y and Y are ϕ-related. For a diffeomorphism ϕ, the push forward is defined, as for forms, by ϕ∗ = (ϕ−1 )∗ . Jacobi–Lie Brackets. If M is finite dimensional and C ∞ then the set of vector fields on M coincides with the set of derivations on F(M ). The same result is true for C k manifolds and vector fields if k ≥ 2. This property is false for infinite-dimensional manifolds; see Abraham, Marsden, Ratiu [1988]. If M is C ∞ and smooth, then the derivation f → X[Y [f ]]−Y [X[f ]], where X[f ] = df · X, determines a unique vector field denoted by [X, Y ] and called the Jacobi–Lie bracket of X and Y . Defining £X Y = [X, Y ] gives the Lie derivative of Y along X. Then the Lie derivative formula (4.3.2) holds with α replaced by Y and the pull back operation given by (4.3.5). If M is infinite-dimensional, then one defines the Lie derivative of Y along X by d ϕ∗ Y = £X Y, t (4.3.7) dt t=0 ........................... 15 July 1998—18h02 ...........................

149. 4.3 The Lie Derivative 135 where ϕt is the flow of X. Then formula (4.3.2) with α replaced by Y holds and the action of the vector field £X Y on a function f is given by X[Y [f ]] − Y [X[f ]] which is denoted, as in the finite-dimensional case, [X, Y ][f ]. As before [X, Y ] = £X Y is also called the Jacobi–Lie bracket of vector fields. If M is finite-dimensional, ∂Y j ∂X j (£X Y )j = X i −Yi = (X · )Y j − (Y · )X j , (4.3.8) ∂xi ∂xi and in general, where we identify X, Y with their local representatives [X, Y ] = DY · X − DX · Y. (4.3.9) The formula for [X, Y ] = £X Y can be remembered by writing ∂ ∂ ∂Y j ∂ ∂X i ∂ Xi ,Y j j = Xi −Yj j . ∂xi ∂x ∂x i ∂xj ∂x ∂xi Algebraic Definiton of the Lie Derivative. The algebraic approach to the Lie derivative on forms or tensors proceeds as follows. Extend the definition of the Lie derivative from functions and vector fields to differential forms, by requiring that the Lie derivative is a derivation; for example, for one-forms α, write £X α, Y = £X α, Y + α, £X Y , (4.3.10) where X, Y are vector fields and α, Y = α(Y ). More generally, k £X (α(Y1 , . . . , Yk )) = (£X α)(Y1 , . . . , Yk ) + α(Y1 , . . . , £X Yi , . . . , Yk ), i=1 (4.3.11) where X, Y1 , . . . , Yk are vector fields and α is a k-form. Proposition 4.3.2. The dynamic and algebraic definitions of the Lie derivative of a differential k-form are equivalent. Cartan’s Magic Formula. A very important formula for the Lie derivative is given by the following. Theorem 4.3.3. For X a vector field and α a k-form on a manifold M , we have £X α = diX α + iX dα, (4.3.12) or, in the “hook” notation, £X α = d(X α) + X dα. ........................... 15 July 1998—18h02 ...........................

150. 136 4. Interlude: Manifolds, Vector Fields, and Differential Forms This is proved by a lengthy but straightforward calculation. Another property of the Lie derivative is the following: if ϕ : M → N is a diffeomorphism, ϕ∗ £Y β = £ϕ∗ Y ϕ∗ β for Y ∈ X(N ), β ∈ Ωk (M ). More generally, if X ∈ X(M ) and Y ∈ X(N ) are ψ related, that is, T ψ ◦ X = Y ◦ ψ for ψ : M → N a smooth map, then £X ψ ∗ β = ψ ∗ £Y β for all β ∈ Ωk (N ). There are a number of valuable identities relating the Lie derivative, the exterior derivative and the interior product which we record at the end of this chapter. For example, if Θ is a one form and X and Y are vector fields, identity 6 in the following table gives dΘ(X, Y ) = X[Θ(Y )] − Y [Θ(X)] − Θ([X, Y ]). (4.3.13) Volume Forms and Divergence. An n-manifold M is said to be orientable if there is a nowhere vanishing n-form µ on it; µ is called a volume form and it is a basis of Ωn (M ) over F(M ). Two volume forms µ1 and µ2 on M are said to define the same orientation if there is an f ∈ F(M ), with f > 0 and such that µ2 = f µ1 . Connected orientable manifolds admit precisely two orientations. A basis {v1 , . . . vn } of Tm M is said to be positively oriented relative to the volume form µ on M if µ(m)(v1 , . . . , vn ) > 0. Note that the volume forms defining the same orientation form a convex cone in Ωn (M ), that is, if a > 0 and µ is a volume form, then aµ is again a volume form and if t ∈ [0, 1] and µ1 , µ2 are volume forms defining the same orientation, then tµ1 + (1 − t)µ2 is again a volume form defining the same orientation as µ1 or µ2 . The first property is obvious. To prove the second, let m ∈ M and let {v1 , . . . vn } be a positively oriented basis of Tm M relative to the orientation defined by µ1 , or equivalently (by hypothesis) by µ2 . Then µ1 (m)(v1 , . . . , vn ) > 0, µ2 (m)(v1 , . . . , vn ) > 0 so that their convex combination is again strictly positive. If µ ∈ Ωn (M ) is a volume form, since £X µ ∈ Ωn (M ), there is a function, called the divergence of X relative to µ and denoted divµ (X) or simply div(X), such that £X µ = divµ (X)µ. (4.3.14) From the dynamic approach to Lie derivatives it follows that divµ (X) = 0 if and only if Ft∗ µ = µ, where Ft is the flow of X. This condition says that Ft is volume preserving . If ϕ : M → M , since ϕ∗ µ ∈ Ωn (M ) there is a function, called the Jacobian of ϕ and denoted Jµ (ϕ) or simply J(ϕ), such that ϕ∗ µ = Jµ (ϕ)µ. (4.3.15) Thus, ϕ is volume preserving if and only if Jµ (ϕ) = 1. From the inverse function theorem, we see that ϕ is a local diffeomorphism if and only if Jµ (ϕ) = 0 on M . ........................... 15 July 1998—18h02 ...........................

151. 4.4 Stokes’ Theorem 137 Frobenius’ Theorem. We also mention a basic result called Frobenius’ theorem. If E ⊂ T M is a vector subbundle, it is said to be involutive if for any two vector fields X, Y on M with values in E, the Jacobi–Lie bracket [X, Y ] is also a vector field with values in E. The subbundle E is said to be integrable if for each point m ∈ M there is a local submanifold of M containing m such that its tangent bundle equals E restricted to this submanifold. If E is integrable, the local integral manifolds can be extended to get, through each m ∈ M , a connected maximal integral manifold, which is unique and is a regularly immersed submanifold of M . The collection of all maximal integral manifolds through all points of M is said to form a foliation. The Frobenius theorem states that the involutivity of E is equivalent to the integrability of E. Exercises 4.3-1. Let M be an n-manifold, µ ∈ Ωn (M ) a volume form, X, Y ∈ X(M ), and f, g : M → R smooth functions such that f (m) = 0 for all m. Prove the following identities: (a) divf µ (X) = divµ (X) + X[f ]/f ; (b) divµ (gX) = g divµ (X) + X[g]; and (c) divµ ([X, Y ]) = X[divµ (Y )] − Y [divµ (X)]. 4.3-2. Show that the partial differential equation n ∂f ∂f = X i (x1 , . . . , xn ) ∂t i=1 ∂xi with initial condition f (x, 0) = g(x) has the solution f (x, t) = g(Ft (x)), where Ft is the flow of the vector field (X 1 , . . . , X n ) in Rn whose flow is assumed to exist for all time. Show that the solution is unique. Generalize this exercise to the equation ∂f = X[f ] ∂t for X a vector field on a manifold M . 4.3-3. Show that if M and N are orientable manifolds, so is M × N . 4.4 Stokes’ Theorem The basic idea of the definition of the integral of an n-form µ on an oriented n-manifold M is to pick a covering by coordinate charts and to sum up the ........................... 15 July 1998—18h02 ...........................

152. 138 4. Interlude: Manifolds, Vector Fields, and Diﬀerential Forms ordinary integrals of f (x1 , . . . , xn ) dx1 · · · dxn , where µ = f (x1 , . . . , xn ) dx1 ∧ · · · ∧ dxn is the local representative of µ, being careful not to count overlaps twice. The change of variables formula guarantees that the result, denoted by M µ, is well deﬁned. If one has an oriented manifold with boundary, then the boundary, ∂M , inherits a compatible orientation. This proceeds in a way that generalizes the relation between the orientation of a surface and its boundary in the classical Stokes’ Theorem in R3 . Theorem 4.4.1. (Stokes’ Theorem) Suppose that M is a compact, oriented k-dimensional manifold with boundary ∂M . Let α be a smooth (k − 1)-form on M . Then dα = α. (4.4.1) M ∂M Special cases of Stokes’ theorem are as follows: The Integral Theorems of Calculus. Stokes’ theorem generalizes and synthesizes the classical theorems: (a) Fundamental Theorem of Calculus. b f (x) dx = f (b) − f (a). (4.4.2) a (b) Green’s Theorem. For a region Ω ⊂ R2 : ∂Q ∂P − dx dy = P dx + Q dy. (4.4.3) Ω ∂x ∂y ∂Ω (c) Divergence Theorem. For a region Ω ⊂ R3 : div F dV = F · n dA. (4.4.4) Ω ∂Ω (d) Classical Stokes’ Theorem. For a surface S ⊂ R3 : ∂R ∂Q − dy ∧ dz S ∂y ∂z ∂P ∂R ∂Q ∂P + − dz ∧ dx + − dx ∧ dy ∂z ∂x ∂x ∂y = n · curl F dA S = P dx + Q dy + R dz, (4.4.5) ∂S where F = (P, Q, R). ........................... 15 July 1998—18h02 ...........................

153. 4.4 Stokes’ Theorem 139 Notice that the Poincar´ lemma generalizes the vector calculus theorems e in R3 saying that if curl F = 0, then F = f and if div F = 0, then F = × G. Recall that it states: If α is closed, then locally α is exact; that is, if dα = 0, then locally α = dβ for some β. On contractible manifolds these statements hold globally. Cohomology. The failure of closed forms to be globally exact leads to the study of a very important topological invariant of M , the de Rham cohomology . The kth de Rham cohomology group, denoted H k (M ), is defined by ker(d : Ωk (M ) → Ωk+1 (M )) H k (M ) := . range (d : Ωk−1 (M ) → Ωk (M )) The de Rham theorem states that these Abelian groups are isomorphic to the so-called singular cohomology groups of M defined in algebraic topology in terms of simplices and that depend only on the topological structure of M and not on its differentiable structure. The isomorphism is provided by integration and the fact that the integration map drops to the preceding quotient is guaranteed by Stokes’ theorem. A useful particular case of this theorem is the following: if M is an orientable compact boundaryless n- manifold, then M µ = 0 if and only if the n-form µ is exact. This statement is equivalent to H n (M ) = R for M compact and orientable. Change of Variables. Another basic result in integration theory is the global change of variables formula. Theorem 4.4.2 (Change of Variables). Let M and N be oriented n- manifolds and let ϕ : M → N be an orientation-preserving diffeomorphism. If α is an n-form on N (with, say, compact support), then ϕ∗ α = α. M N Identities for Vector Fields and Forms 1. Vector fields on M with the bracket [X, Y ] form a Lie algebra; that is, [X, Y ] is real bilinear, skew-symmetric, and Jacobi’s identity holds: [[X, Y ], Z] + [[Z, X], Y ] + [[Y, Z], X] = 0. Locally, [X, Y ] = DY · X − DX · Y = (X · )Y − (Y · )X ........................... 15 July 1998—18h02 ...........................

154. 140 4. Interlude: Manifolds, Vector Fields, and Differential Forms and on functions, [X, Y ][f ] = X[Y [f ]] − Y [X[f ]]. 2. For diffeomorphisms ϕ and ψ, ϕ∗ [X, Y ] = [ϕ∗ X, ϕ∗ Y ] and (ϕ ◦ ψ)∗ X = ϕ∗ ψ∗ X. 3. The forms on a manifold comprise a real associative algebra with ∧ as multiplication. Furthermore, α ∧ β = (−1)kl β ∧ α for k and l-forms α and β, respectively. 4. For maps ϕ and ψ, ϕ∗ (α ∧ β) = ϕ∗ α ∧ ϕ∗ β and (ϕ ◦ ψ)∗ α = ψ ∗ ϕ∗ α. 5. d is a real linear map on forms, ddα = 0, and d(α ∧ β) = dα ∧ β + (−1)k α ∧ dβ for α a k-form. 6. For α a k-form and X0 , . . . , Xk vector fields, k (dα)(X0 , . . . , Xk ) = ˆ (−1)i Xi [α(X0 , . . . , Xi , . . . , Xk )] i=0 + (−1) i+j ˆ ˆ α([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xk ) 0≤i<j≤k ˆ where Xi means that Xi is omitted. Locally, k dα(x)(v0 , . . . , vk ) = (−1)i Dα(x) · vi (v0 , . . . , vi , . . . , vk ). ˆ i=0 7. For a map ϕ, ϕ∗ dα = dϕ∗ α. 8. Poincar´ Lemma. If dα = 0, then the k-form α is locally exact; e that is, there is a neighborhood U about each point on which α = dβ. This statement is global on contractible manifolds. 9. iX α is real bilinear in X, α and for h : M → R, ihX α = hiX α = iX hα. Also, iX iX α = 0 and iX (α ∧ β) = iX α ∧ β + (−1)k α ∧ iX β for α a k-form. ........................... 15 July 1998—18h02 ...........................

155. 4.4 Stokes’ Theorem 141 10. For a diﬀeomorphism ϕ, ϕ∗ (iX α) = iϕ∗ X (ϕ∗ α); i.e., ϕ∗ (X α) = (ϕ∗ X) (ϕ∗ α) if f : M → N is a mapping and Y is f -related to X, that is, T f ◦ X = Y ◦ f, then iX f ∗ α = f ∗ iY α; i.e., X (f ∗ α) = f ∗ (Y α). 11. £X α is real bilinear in X, α and £X (α ∧ β) = £X α ∧ β + α ∧ £X β. 12. Cartan’s Magic Formula: £X α = diX α + iX dα = d(X α) + X dα 13. For a diﬀeomorphism ϕ, ϕ∗ £X α = £ϕ∗ X ϕ∗ α; if f : M → N is a mapping and Y is f -related to X, then £Y f ∗ α = f ∗ £X α. 14. (£X α)(X1 , . . . , Xk ) = X[α(X1 , . . . , Xk )] k − α(X1 , . . . , [X, Xi ], . . . , Xk ). i=0 Locally, (£X α)(x) · (v1 , . . . , vk ) = (Dαx · X(x))(v1 , . . . , vk ) k + αx (v1 , . . . , DXx · vi , . . . , vk ). i=0 15. The following identities hold: (a) £f X α = f £X α + df ∧ iX α; (b) £[X,Y ] α = £X £Y α − £Y £X α; (c) i[X,Y ] α = £X iY α − iY £X α; (d) £X dα = d£X α; and (e) £X iX α = iX £X α. ........................... 15 July 1998—18h02 ...........................

156. 142 4. Interlude: Manifolds, Vector Fields, and Differential Forms (f) £X (α ∧ β) = £X α ∧ β + α ∧ £X β 16. If M is a finite-dimensional manifold, X = X l ∂/∂xl , and α = αi1 ...ik dxi1 ∧ · · · ∧ dxik , where i1 < · · · < ik , then the following formulas hold: ∂αi1 ...ik dα = dxl ∧ dxi1 ∧ · · · ∧ dxik , ∂xl iX α = X l αli2 ...ik dxi2 ∧ · · · ∧ dxik , ∂αi1 ...ik £X α = X l dxi1 ∧ · · · ∧ dxik ∂xl ∂X l + αli2 ...ik dxi1 ∧ dxi2 ∧ . . . ∧ dxik + . . . . ∂xi1 Exercises 4.4-1. Let Ω be a closed bounded region in R2 . Use Green’s theorem to show that the area of Ω equals the line integral 1 (x dy − y dx). 2 ∂Ω 4.4-2. On R2 {(0, 0)} consider the one-form α = (x dy − y dx)/(x2 + y 2 ). (a) Show that this form is closed. (b) Using the angle θ as a variable on S 1 , compute i∗ α, where i : S 1 → R2 is the standard embedding. (c) Show that α is not exact. 4.4-3. The magnetic monopole Let B = gr/r3 be a vector field on Euclidean three-space minus the origin where r = r . Show that B cannot be written as the curl of something. ........................... 15 July 1998—18h02 ...........................

157. This is page 143 Printer: Opaque this 5 Hamiltonian Systems on Symplectic Manifolds Now we are ready to geometrize Hamiltonian mechanics to the context of manifolds. First we make phase spaces nonlinear and then we study Hamiltonian systems in this context. 5.1 Symplectic Manifolds Definition 5.1.1. A symplectic manifold is a pair (P, Ω), where P is a manifold and Ω is a closed (weakly) nondegenerate two-form on P . If Ω is strongly nondegenerate, we speak of a strong symplectic manifold . As in the linear case, strong nondegeneracy of the two-form Ω means that at each z ∈ P, the bilinear form Ωz : Tz P × Tz P → R is nondegenerate, that is, Ωz defines an isomorphism ∗ Ωz : Tz P → Tz P. For a (weak) symplectic form, the induced map Ω : X(P ) → X∗ (P ) between vector fields and one-forms is one-to-one, but in general is not surjective. We will see later that Ω is required to be closed, that is, dΩ = 0, where d is the exterior derivative, so that the induced Poisson bracket satisfies the Jacobi identity and so that the flows of Hamiltonian vector fields will consist of canonical transformations. In coordinates z I on P in the finite-dimensional case, if Ω = ΩIJ dz I ∧ dz J (sum over all I < J), then

158. 144 5. Hamiltonian Systems on Symplectic Manifolds dΩ = 0 becomes the condition ∂ΩIJ ∂ΩKI ∂ΩJK + + = 0. (5.1.1) ∂z K ∂z J ∂z I Examples (a) Symplectic Vector Spaces. If (Z, Ω) is a symplectic vector space, then it is also a symplectic manifold. The requirement dΩ = 0 is satisfied automatically since Ω is a constant form (that is, Ω(z) is independent of z ∈ Z). (b) The cylinder S 1 × R with coordinates (θ, p) is a symplectic manifold with Ω = dθ ∧ dp. (c) The torus T2 with periodic coordinates (θ, ϕ) is a symplectic manifold with Ω = dθ ∧ dϕ. (d) The two-sphere S 2 of radius r is symplectic with Ω the standard area element Ω = r2 sin θ dθ ∧ dϕ on the sphere as the symplectic form. Given a manifold Q, we will show in Chapter 6 that the cotangent bundle T ∗ Q has a natural symplectic structure. When Q is the configuration space of a mechanical system, T ∗ Q is called the momentum phase space. This important example generalizes the linear examples with phase spaces of the form W × W ∗ that we studied in Chapter 2. Darboux’ Theorem. The next result says that, in principle, every strong symplectic manifold is, in suitable local coordinates, a symplectic vector space. (By contrast, a corresponding result for Riemannian manifolds is not true unless they have zero curvature; that is, are flat.) Theorem 5.1.2 (Darboux’ Theorem). Let (P, Ω) be a strong symplectic manifold. Then in a neighborhood of each z ∈ P , there is a local coordinate chart in which Ω is constant. Proof. We can assume P = E and z = 0 ∈ E, where E is a Banach space. Let Ω1 be the constant form equaling Ω(0). Let Ω = Ω1 − Ω and Ωt = Ω + tΩ , for 0 ≤ t ≤ 1. For each t, the bilinear form Ωt (0) = Ω(0) is nondegenerate. Hence by openness of the set of linear isomorphisms of E to E ∗ and compactness of [0, 1], there is a neighborhood of 0 on which Ωt is strongly nondegenerate for all 0 ≤ t ≤ 1. We can assume that this neighborhood is a ball. Thus by the Poincar´ lemma, Ω = dα for some e one-form α. Replacing α by α − α(0), we can suppose α(0) = 0. Define a smooth time-dependent vector field Xt by iXt Ωt = −α, ........................... 15 July 1998—18h02 ...........................

159. 5.1 Symplectic Manifolds 145 which is possible since Ωt is strongly nondegenerate. Since α(0) = 0 we get Xt (0) = 0, and so from the local existence theory for ordinary differential equations, there is a ball on which the integral curves of Xt are defined for a time at least one; see Abraham, Marsden, and Ratiu [1988], §4.1, for the technical theorem. Let Ft be the flow of Xt starting at F0 = identity. By the Lie derivative formula for time-dependent vector fields, we have d ∗ d (Ft Ωt ) = Ft∗ (£Xt Ωt ) + Ft∗ Ωt dt dt = Ft∗ diXt Ωt + Ft∗ Ω = Ft∗ (d(−α) + Ω ) = 0. ∗ ∗ Thus, F1 Ω1 = F0 Ω0 = Ω, so F1 provides a chart transforming Ω to the constant form Ω1 . This proof is due to Moser [1965]. As was noted by Weinstein [1971], this proof generalizes to the infinite-dimensional strong symplectic case. Unfor- tunately, many interesting infinite-dimensional symplectic manifolds are not strong. In fact, the analog of Darboux’s theorem is not valid for weak symplectic forms. For an example, see Exercise 5.1-3, and for conditions under which it is valid, see Marsden [1981], Olver [1988], Bambusi [1998], and references therein. For an equivariant Darboux theorem and references, see Dellnitz and Melbourne [1993], and the discussion in Chapter 9. Corollary 5.1.3. If (P, Ω) is a finite-dimensional symplectic manifold, then P is even dimensional, and in a neighborhood of z ∈ P there are local coordinates (q 1 , . . . , q n , p1 , . . . , pn ) (where dim P = 2n) such that n Ω= dq i ∧ dpi . (5.1.2) i=1 This follows from Darboux’s theorem and the canonical form for linear symplectic forms. As in the vector space case, coordinates in which Ω takes the above form are called canonical coordinates. Corollary 5.1.4. If (P, Ω) is a 2n-dimensional symplectic manifold, then P is oriented by the Liouville volume (−1)n(n−1)/2 Λ= Ω ∧ ··· ∧ Ω (n times). (5.1.3) n! In canonical coordinates (q 1 , . . . , q n , p1 , . . . , pn ), Λ has the expression Λ = dq 1 ∧ · · · ∧ dq n ∧ dp1 ∧ · · · ∧ dpn . (5.1.4) Thus, if (P, Ω) is a 2n-dimensional symplectic manifold, then (P, Λ) is a volume manifold (that is, a manifold with a volume element). The measure associated to Λ is called the Liouville measure. The factor (−1)n(n−1)/2 /n! is chosen so that in canonical coordinates, Λ has the expression (5.1.4). ........................... 15 July 1998—18h02 ...........................

160. 146 5. Hamiltonian Systems on Symplectic Manifolds Exercises 5.1-1. Show how to construct (explicitly) canonical coordinates for the symplectic form Ω = f µ on S 2 , where µ is the standard area element and where f : S 2 → R is a positive function. 5.1-2. (Moser [1965]). Let µ0 and µ1 be two volume elements (nowhere vanishing n-forms) on the compact boundaryless n-manifold M giving M the same orientation. Assume that M µ0 = M µ1 . Show that there is a diffeomorphism ϕ : M → M such that ϕ∗ µ1 = µ0 . 5.1-3. (Requires some functional analysis) Prove that Darboux’ theorem fails for the following weak symplectic form. Let H be a real Hilbert space and S : H → H a compact, self-adjoint, and positive operator whose range is dense in H, but not equal to H. Let Ax = S + x 2 I and gx (e, f ) = Ax e, f . Let Ω be the weak symplectic form on H × H associated to g. Show that there is no coordinate chart about (0, 0) ∈ H × H on which Ω is constant. 5.1-4. Use the method of proof of the Darboux Theorem to show the following. Assume that Ω0 and Ω1 are two symplectic forms on the compact manifold P such that [Ω0 ], [Ω1 ] are the cohomology classes of Ω0 and Ω1 respectively in H 2 (P ; R). If for every t ∈ [0, 1], the form Ωt := (1−t)Ω0 +Ω1 is non-degenerate, show that there is a diffeomorphism ϕ : P −→ P such that ϕ∗ Ω1 = Ω0 . 5.1-5. Prove the following Relative Darboux Theorem. Let S be a submanifold of P and assume that Ω0 and Ω1 are two strong symplectic forms on P such that Ω0 |S = Ω1 |S. Then there is an open neighborhood V of S in P and a diffeomorphism ϕ : V −→ ϕ(V ) ⊂ P such that ϕ|S = identity on S and ϕ∗ Ω1 = Ω0 . (Hint: Use Exercise 4.2-6.) Check solution 5.2 Symplectic Transformations Definition 5.2.1. Let (P1 , Ω1 ) and (P2 , Ω2 ) be symplectic manifolds. A C ∞ -mapping ϕ : P1 → P2 is called symplectic or canonical if ϕ∗ Ω2 = Ω1 . (5.2.1) Recall that Ω1 = ϕ∗ Ω2 means that for each z ∈ P1 , and all v, w ∈ Tz P1 , we have the following identity: Ω1z (v, w) = Ω2ϕ(z) (Tz ϕ · v, Tz ϕ · w), where Ω1z means Ω1 evaluated at the point z and where Tz ϕ is the tangent (derivative) of ϕ at z. ........................... 15 July 1998—18h02 ...........................

161. 5.2 Symplectic Transformations 147 If ϕ : (P1 , Ω1 ) → (P2 , Ω2 ) is canonical, the property ϕ∗ (α ∧ β) = ϕ∗ α ∧ ϕ β implies that ϕ∗ Λ = Λ; that is, ϕ also preserves the Liouville measure. ∗ Thus we get the following: Proposition 5.2.2. A smooth canonical transformation between symplectic manifolds of the same dimension is volume preserving and is a local diffeomorphism. The last statement comes from the inverse function theorem: if ϕ is volume preserving, its Jacobian determinant is 1, so ϕ is locally invertible. It is clear that the set of canonical diffeomorphisms of P form a subgroup of Diff(P ), the group of all diffeomorphisms of P . This group, denoted Diff can (P ), plays a key role in the study of plasma dynamics. If Ω1 and Ω2 are exact, say Ω1 = −dΘ1 and Ω2 = −dΘ2 , then (5.2.1) is equivalent to d(ϕ∗ Θ2 − Θ1 ) = 0. (5.2.2) Let M ⊂ P1 be an oriented two-manifold with boundary ∂M . Then if (5.2.2) holds, we get 0= d(ϕ∗ Θ2 − Θ1 ) = (ϕ∗ Θ2 − Θ1 ) , M ∂M that is, ϕ∗ Θ2 = Θ1 . (5.2.3) ∂M ∂M Proposition 5.2.3. The map ϕ : P1 → P2 is canonical if and only if (5.2.3) holds for every oriented two-manifold M ⊂ P1 with boundary ∂M . The converse is proved by choosing M to be a small disk in P1 and using the statement: if the integral of a two-form over any small disk vanishes, then the form is zero. The latter assertion is proved by contradiction, constructing a two-form on a two-disk whose coefficient is a bump function. Equation (5.2.3) is an example of an integral invariant. For more information, see Arnold [1989] and Abraham and Marsden [1978]. Exercises 5.2-1. Let ϕ : R2n → R2n be a map of the form ϕ(q, p) = (q, p + α(q)). Use the canonical one-form to determine when ϕ is symplectic. 5.2-2. Let T6 be the six-torus with symplectic form Ω = dθ1 ∧ dθ2 + dθ3 ∧ dθ4 + dθ5 ∧ dθ6 . ........................... 15 July 1998—18h02 ...........................

162. 148 5. Hamiltonian Systems on Symplectic Manifolds Show that if ϕ : T6 → T6 is symplectic and M ⊂ T6 is a compact oriented four-manifold with boundary, then ϕ∗ (Ω ∧ Θ) = Ω ∧ Θ, ∂M ∂M where Θ = θ1 dθ2 + θ3 dθ4 + θ5 dθ6 . 5.2-3. Show that any canonical map between finite-dimensional symplectic manifolds is an immersion. 5.3 Complex Structures and K¨hler a Manifolds This section develops the relation between complex and symplectic geometry a little further. It may be omitted on a first reading. Complex Structures. We begin with the case of vector spaces. By a complex structure on a real vector space Z, we mean a linear map J : Z → Z such that J2 = −Identity. Setting iz = J(z) gives Z the structure of a complex vector space. Note that if Z is finite dimensional, the hypothesis on J implies that (det J)2 = (−1)dim Z , so dim Z must be an even number since det J ∈ R. The complex dimension of Z is half the real dimension. Conversely, if Z is a complex vector space, it is also a real vector space by restricting scalar multiplication to the real numbers. In this case, Jz = iz is the complex structure on Z. As before, the real dimension of Z is twice the complex dimension since the vectors z and iz are linearly independent. We have already seen that the imaginary part of a complex inner product is a symplectic form. Conversely, if H is a real Hilbert space and Ω is a skew-symmetric weakly nondegenerate bilinear form on H, then there is a complex structure J on H and a real inner product s such that s(z, w) = −Ω(Jz, w). (5.3.1) The expression h(z, w) = s(z, w) − iΩ(z, w) (5.3.2) defines a Hermitian inner product, and h or s is complete on H if and only if Ω is strongly nondegenerate. (See Abraham and Marsden [1978], p.173, for the proof.) Moreover, given any two of (s, J, Ω), there is at most one third structure such that (5.3.1) holds. If we identify Cn with R2n and write z = (z1 , . . . , zn ) = (x1 + iy1 , . . . , xn + iyn ) = ((x1 , y1 ), . . . , (xn , yn )), ........................... 15 July 1998—18h02 ...........................

163. 5.3 Complex Structures and K¨hler Manifolds a 149 then − Im (z1 , . . . , zn ), (z1 , . . . , zn ) = − Im(z1 z 1 + · · · + zn z n ) = −(x1 y1 − x1 y1 + · · · + xn yn − xn yn ). Thus, the canonical symplectic form on R2n may be written Ω(z, z ) = − Im z, z = Re iz, z , (5.3.3) which, by (5.3.1), agrees with the convention that J : R2n → R2n is multiplication by i. An almost complex stucture J on a manifold M is a smooth tangent bundle isomorphism J : T M → T M covering the identity map on M such that for each point z ∈ M , Jz = J(z) : Tz M → Tz M is a complex structure on the vector space Tz M . A manifold with an almost complex structure is called an almost complex manifold . A manifold M is called a complex manifold if it admits an atlas {(Uα , ϕα )} whose charts ϕα : Uα ⊂ M → E map to a complex Banach space E and the transition functions ϕβ ◦ϕ−1 : ϕα (Uα ∩Uβ ) → ϕβ (Uα ∩Uβ ) α are holomorphic maps. The complex structure on E (multiplication by i) induces via the chart maps ϕα an almost complex structure on each chart domain Uα . Since the transition functions are biholomorphic diffeomorphisms, the almost complex structures on Uα ∩ Uβ induced by ϕα and ϕβ coincide. This shows that a complex manifold is also almost complex. The converse is not true. If M is an almost complex manifold, Tz M is endowed with the structure of a complex vector space. A Hermitian metric on M is a smooth assignment of a (possibly weak) complex inner product on Tz M for each z ∈ M . As in the case of vector spaces, the imaginary part of the Hermitian metric defines a non-degenerate (real) two-form on M . The real part of a Hermitian metric is a Riemannian metric on M . If the complex inner product on each tangent space is strongly nondegenerate, the metric is strong ; in this case both the real and imaginary parts of the Hermitian metric are strongly nondegenerate over R. K¨hler Manifolds. An almost complex manifold M with a Hermitian a metric , is called a K¨hler manifold , if M is a complex manifold and a the two-form − Im , is a closed two form on M . There is an equivalent definition that is often useful: A K¨hler manifold is a smooth manifold a with a Riemannian metric g and an almost complex structure J such that Jz is g-skew for each z ∈ M and such that J is covariantly constant with respect to g. (One requires some Riemannian geometry to understand this definition—it will not be required in what follows.) The important fact used later on is the following: ........................... 15 July 1998—18h02 ...........................

164. 150 5. Hamiltonian Systems on Symplectic Manifolds Any K¨hler manifold is also symplectic, with symplectic form a given by Ωz (vz , wz ) = Jz vz , wz . (5.3.4) In this second definition of K¨hler manifolds, the condition dΩ = 0 follows a from J being covariantly constant. A strong K¨hler manifold is a K¨hler a a manifold whose Hermitian inner product is strong. Projective Spaces. Any complex Hilbert space H is a strong K¨hler a manifold. As an example of a more interesting K¨hler manifold, we shall a consider the projectivization PH of a complex Hilbert space H. In particular, complex projective n-space CPn will result when this construction is applied to Cn . Recall from Example (f) of §2.3 that H is a symplectic vector space relative to the quantum mechanical symplectic form Ω(ψ1 , ψ2 ) = −2 Im ψ1 , ψ2 , where , is the Hermitian inner product on H, is Planck’s constant, and ψ1 , ψ2 ∈ H. Recall also that PH is the space of complex lines through the origin in H. Denote by π : H{0} → PH the canonical projection which sends a vector ψ ∈ H{0} to the complex line it spans, denoted by [ψ] when thought of as a point in PH and by Cψ when interpreted as a subspace of H. The space PH is a smooth complex manifold, π is a smooth map, and the tangent space T[ψ] PH is isomorphic to H/Cψ. Thus, the map π is a surjective submersion. (See Abraham, Marsden, Ratiu [1988], Chapter 3.) Since the kernel of Tψ π : H → T[ψ] PH is Cψ, the map Tψ π|(Cψ)⊥ is a complex linear isomorphism from (Cψ)⊥ to Tψ PH that depends on the chosen representative ψ in [ψ]. If U : H → H is a unitary operator, that is, U is invertible and U ψ1 , U ψ2 = ψ1 , ψ2 for all ψ1 , ψ2 ∈ H, then the rule [U ][ψ] := [U ψ] defines a biholomorphic diffeomorphism on PH. Proposition 5.3.1. (i) If [ψ] ∈ PH, ψ = 1, and ϕ1 , ϕ2 ∈ (Cψ)⊥ , the formula Tψ π(ϕ1 ), Tψ π(ϕ2 ) = 2 ϕ1 , ϕ2 (5.3.5) gives a well-defined strong Hermitian inner product on T[ψ] PH, that is, the left hand side does not depend on the choice of ψ in [ψ]. The dependence on [ψ] is smooth and so (5.3.5) defines a Hermitian metric on PH called the Fubini-Study metric. This metric is invariant under the action of the maps [U ], for all unitary operators U on H. ........................... 15 July 1998—18h02 ...........................

165. 5.3 Complex Structures and K¨hler Manifolds a 151 (ii) For [ψ] ∈ PH, ψ = 1, and ϕ1 , ϕ2 ∈ (Cψ)⊥ , g[ψ] (Tψ π(ϕ1 ), Tψ π(ϕ2 )) = 2 Re ϕ1 , ϕ2 (5.3.6) defines a strong Riemannian metric on PH invariant under all transformations [U ]. (iii) For [ψ] ∈ PH, ψ = 1, and ϕ1 , ϕ2 ∈ (Cψ)⊥ , Ω[ψ] (Tψ π(ϕ1 ), Tψ π(ϕ2 )) = −2 Im ϕ1 , ϕ2 (5.3.7) defines a strong symplectic form on PH invariant under all transformations [U ]. Proof. We first prove1 (i). If λ ∈ C{0}, then π(λ(ψ + tϕ)) = π(ψ + tϕ), and since d d (Tλψ π)(λϕ) = π(λψ + tλϕ) = π(ψ + tϕ) = (Tψ π)(ϕ), dt t=0 dt t=0 we get (Tλψ π)(λϕ) = (Tψ π)(ϕ). Thus, if λψ = ψ = 1, it follows that |λ| = 1. We have by (5.3.5), (Tλψ π)(λϕ1 ), (Tλψ π)(λϕ2 ) = 2 λϕ1 , λϕ2 = 2 |λ|2 ϕ1 , ϕ2 =2 ϕ1 , ϕ2 = (Tψ π)(ϕ1 ), (Tψ π)(ϕ2 ) . This shows that the definition (5.3.5) of the Hermitian inner product is independent on the normalized representative ψ ∈ [ψ] chosen in order to define it. This Hermitian inner product is strong since it coincides with the inner product on the complex Hilbert space (Cψ)⊥ . A straightforward computation (see Exercise 5.3-3) shows that for ψ ∈ H{0} and ϕ1 , ϕ2 ∈ H arbitrary, the Hermitian metric is given by −2 −2 Tψ π(ϕ1 ), Tψ π(ϕ2 ) = 2 ψ ( ϕ1 , ϕ2 − ψ ϕ1 , ψ ψ, ϕ2 ). (5.3.8) Since the right hand side is smooth in ψ ∈ H{0} and this formula drops to PH, it follows that (5.3.5) is smooth in [ψ]. If U is a unitary map on H and [U ] is the induced map on PH, we have d d T[ψ] [U ] · Tψ π(ϕ) = T[ψ] [U ] · [ψ + tϕ] = [U ][ψ + tϕ] dt t=0 dt t=0 d = [U (ψ + tϕ)] = TU ψ π(U ϕ). dt t=0 1 One can give a conceptually cleaner, but more advanced approach to this process using general reduction theory. The proof given here is by a direct argument. ........................... 15 July 1998—18h02 ...........................

166. 152 5. Hamiltonian Systems on Symplectic Manifolds Therefore, since U ψ = ψ = 1 and U ϕj , U ψ = 0, we get by (5.3.5), T[ψ] [U ] · Tψ π(ϕ1 ), T[ψ] [U ] · Tψ π(ϕ2 ) = TU ψ π(U ϕ1 ), TU ψ π(U ϕ2 ) = U ϕ1 , U ϕ2 = ϕ1 , ϕ2 = Tψ π(ϕ1 ), Tψ π(ϕ2 ) , which proves the invariance of the Hermitian metric under the action of the transformation [U ]. Part (ii) is obvious as the real part of the Hermitian metric (5.3.5). Finally we prove (iii). From the invariance of the metric it follows that the form Ω is also invariant under the action of unitary maps, that is, [U ]∗ Ω = Ω. So, also [U ]∗ dΩ = dΩ. Now consider the unitary map U0 on H defined by U0 ψ = ψ and U0 = −Identity on (Cψ)⊥ . Then from [U0 ]∗ Ω = Ω we have for ϕ1 , ϕ2 , ϕ3 ∈ (Cψ)⊥ dΩ([ψ])(Tψ π(ϕ1 ), Tψ π(ϕ2 ), Tψ π(ϕ3 )) = dΩ([ψ])(T[ψ] [U0 ] · Tψ π(ϕ1 ), T[ψ] [U0 ] · Tψ π(ϕ2 ), T[ψ] [U0 ] · Tψ π(ϕ3 )). But T[ψ] [U0 ] · Tψ π(ϕ) = Tψ π(−ϕ) = −Tψ π(ϕ), which implies by trilinearity of dΩ that dΩ = 0. The symplectic form Ω is strongly nondegenerate since on T[ψ] PH it restricts to the corresponding quantum mechanical symplectic form on the Hilbert space (Cψ)⊥ . The results above prove that PH is an infinite dimensional K¨hler man- a ifold on which the unitary group U (H) acts by isometries. This can be generalized to Grassmannian manifolds of finite (or infinite) dimensional subspaces of H, and even more, to flag manifolds (see Besse [1987], Pressley and Segal [1985]). Exercises 5.3-1. On Cn , show that Ω = −dΘ, where Θ(z) · w = 1 2 Im z, w . 5.3-2. Let P be a manifold that is both symplectic, with symplectic form Ω and is Riemannian, with metric g. (a) Show that P has an almost complex structure J such that Ω(u, v) = g(Ju, v) if and only if Ω( F, v) = −g(XF , v) for all F ∈ F(P ). (b) Under the hypothesis of (a), show that a Hamiltonian vector field XH is locally a gradient if and only if £ H Ω = 0. ........................... 15 July 1998—18h02 ...........................

167. 5.4 Hamiltonian Systems 153 5.3-3. Show that for any vectors ϕ1 , ϕ2 ∈ H and ψ = 0 the Fubini-Study metric can be written: −2 −2 Tψ π(ϕ1 ), Tψ π(ϕ2 ) = 2 ψ ( ϕ1 , ϕ2 − ψ ϕ1 , ψ ψ, ϕ2 ). Conclude that the Riemannian metric and symplectic forms are given by 2 g[ψ] (Tψ π(ϕ1 ), Tψ π(ϕ2 )) = 4 Re( ϕ1 , ϕ2 ψ 2 − ϕ1 , ψ ψ, ϕ2 ) ψ and 2 Ω[ψ] (Tψ π(ϕ1 ), Tψ π(ϕ2 )) = − 4 Im( ϕ1 , ϕ2 ψ 2 − ϕ1 , ψ ψ, ϕ2 ). ψ 5.3-4. Prove that dΩ = 0 on PH directly without using the invariance under the maps [U ], for U a unitary operator on H. 5.3-5. For Cn+1 , show that in a projective chart of CPn the symplectic form Ω is determined by: π ∗ Ω = (1 + |z|2 )−1 (dσ − (1 + |z|2 )−1 σ ∧ σ), n+1 where d|z|2 = σ + σ (explicitly, σ = i=1 zi dz i ) and π : Cn {0} → CPn is the projection.. Use this to show that dΩ = 0. Note the similarity between this formula and the corresponding one in 5.3-3. 5.4 Hamiltonian Systems Definition 5.4.1. Let (P, Ω) be a symplectic manifold. A vector field X on P is called Hamiltonian if there is a function H : P → R such that iX Ω = dH; (5.4.1) that is, for all v ∈ Tz P , we have the identity Ωz (X(z), v) = dH(z) · v. In this case we write XH for X. The set of all Hamiltonian vector fields on P is denoted XHam (P ). Hamilton’s equations are the evolution equations z = XH (z). ˙ In finite dimensions, Hamilton’s equations in canonical coordinates are dq i ∂H dpi ∂H = , =− i. dt ∂pi dt ∂q ........................... 15 July 1998—18h02 ...........................

168. 154 5. Hamiltonian Systems on Symplectic Manifolds Vector Fields and Flows. A vector field X is called locally Hamilto- nian if iX Ω is closed. This is equivalent to £X Ω = 0, where £X Ω denotes Lie differentiation of Ω along X, because £X Ω = iX dΩ + diX Ω = diX Ω. If X is locally Hamiltonian, it follows from the Poincar´ lemma that there e locally exists a function H such that iX Ω = dH, so locally X = XH and thus the terminology is consistent. In a symplectic vector space, we have seen in Chapter 2 that the condition that iX Ω be closed is equivalent to DX(z) being Ω-skey. Thus, the definition of locally Hamiltonian is an intrinsic generalization of what we did in the vector space case. The flow ϕt of a locally Hamiltonian vector field X satisfies ϕ∗ Ω = Ω t since d ∗ ϕ Ω = ϕ∗ £X Ω = 0, dt t t and thus we have proved the following: Proposition 5.4.2. The flow ϕt of a vector field X consists of symplectic transformations (that is, for each t, ϕ∗ Ω = Ω where defined) if and only if t X is locally Hamiltonian. A constant vector field on the torus T2 gives an example of a locally Hamiltonian vector field that is not Hamiltonian. (See Exercise 5.4-1.) Energy Conservation. If XH is Hamiltonian with flow ϕt , then by the chain rule, d (Hϕt (z)) = dH(ϕt (z)) · XH (ϕt (z)) dt = Ω (XH (ϕt (z)), XH (ϕt (z))) = 0, (5.4.2) since Ω is skew. Thus H ◦ ϕt is constant in t. We have proved the following: Proposition 5.4.3 (Conservation of Energy). If ϕt is the flow of XH on the symplectic manifold P , then H ◦ ϕt = H (where defined ). Transformation of Hamiltonian Systems. As in the vector space case, we have: Proposition 5.4.4. A diffeomorphism ϕ : P1 → P2 of symplectic manifolds is symplectic if and only if it satisfies ϕ∗ XH = XH◦ϕ (5.4.3) for all functions H : U → R (such that XH is defined) where U is any open subset of P2 . ........................... 15 July 1998—18h02 ...........................

169. 5.4 Hamiltonian Systems 155 Proof. The statement (5.4.3) means that for each z ∈ P , Tϕ(z) ϕ−1 · XH (ϕ(z)) = XH◦ϕ (z) that is, XH (ϕ(z)) = Tz ϕ · XH◦ϕ (z). In other words, Ω(ϕ(z))(XH (ϕ(z)), Tz ϕ · v) = Ω(ϕ(z))(Tz ϕ · XH◦ϕ (z), Tz ϕ · v) for all v ∈ Tz P . If ϕ is symplectic, this becomes dH(ϕ(z)) · [Tz ϕ · v] = d(H ◦ ϕ)(z) · v, which is true by the chain rule. Thus, if ϕ is symplectic, then (5.4.3) holds. The converse is proved in the same way. The same qualifications on technicalities pertinent to the infinite-dimensional case that were discussed for vector spaces apply to the present context as well. For instance, given H, there is no a priori guarantee that XH exists: we usually assume it abstractly and verify it in examples. Also, we may wish to deal with XH ’s that have dense domains rather than everywhere defined smooth vector fields. These technicalities are important, but do not affect many of the main goals of this book. We shall, for simplicity, deal only with everywhere defined vector fields and refer the reader to Chernoff and Marsden [1974] and Marsden and Hughes [1983] for the general case. We shall also tacitly restrict our attention to functions which have Hamiltonian vector fields. Of course in the finite-dimensional case these technical problems disappear. Exercises 5.4-1. Let X be a constant nonzero vector field on the two-torus. Show that X is locally Hamiltonian but is not globally Hamiltonian. 5.4-2. Show that the bracket of two locally Hamiltonian vector fields on a symplectic manifold (P, Ω) is globally Hamiltonian. 5.4-3. Consider the equations on C2 given by z1 = −iw1 z1 + ipz 2 + iz1 (a|z1 |2 + b|z2 |2 ), ˙ z2 = −iw2 z2 + iqz 1 + iz2 (c|z1 |2 + d|z2 |2 ). ˙ Show that this system is Hamiltonian if and only if p = q and b = c with 1 a b d H= w2 |z2 |2 + w1 |z1 |2 − p Re(z1 z2 ) − |z1 |4 − |z1 z2 |2 − |z2 |4 . 2 4 2 4 ........................... 15 July 1998—18h02 ...........................

170. 156 5. Hamiltonian Systems on Symplectic Manifolds 5.4-4. Let (P, Ω) be a symplectic manifold and ϕ : S −→ P an immersion. ϕ is called a coisotropic immersion if Ts ϕ(Ts S) is a coisotropic subspace of Tϕ(s) P for every s ∈ S. This means that [Ts ϕ(Ts S)]Ω(s) ⊂ Ts ϕ(Ts S) for every s ∈ S (see Exercise 2.3-5). If (P, Ω) is a strong symplectic manifold, show that ϕ : S −→ P is a coisotropic immersion if and only if XH (ϕ(s)) ∈ Ts ϕ(Ts S) for all s ∈ S, all open neighborhoods U of ϕ(s) in P , and all smooth functions H : U −→ R satisfying H|ϕ(S) ∩ U = constant 5.5 Poisson Brackets on Symplectic Manifolds Analogous to the vector space treatment, we define the Poisson bracket of two functions F, G : P → R by {F, G}(z) = Ω(XF (z), XG (z)). (5.5.1) From Proposition 5.4.4 we get (see the proof of Proposition 2.7.5): Proposition 5.5.1. A diffeomorphism ϕ : P1 → P2 is symplectic if and only if {F, G} ◦ ϕ = {F ◦ ϕ, G ◦ ϕ} (5.5.2) for all functions F, G ∈ F(U ), where U is an arbitrary open subset of P2 . Using this, Proposition 5.4.2 shows that Proposition 5.5.2. If ϕt is the flow of a Hamiltonian vector field XH (or a locally Hamiltonian vector field), then ϕ∗ {F, G} = {ϕ∗ F, ϕ∗ G} t t t for all F, G ∈ F(P ) (or restricted to an open set if the flow is not everywhere defined). Corollary 5.5.3. The following derivation identity holds: XH [{F, G}] = {XH [F ], G} + {F, XH [G]} (5.5.3) where we use the notation XH [F ] = £XH F for the derivative of F in the direction XH . Proof. Differentiate the identity ϕ∗ {F, G} = {ϕ∗ F, ϕ∗ G} t t t ........................... 15 July 1998—18h02 ...........................

171. 5.5 Poisson Brackets on Symplectic Manifolds 157 in t at t = 0, where ϕt is the flow of XH . The left-hand side clearly gives the left side of (5.5.3). To evaluate the right-hand side, first notice that d d Ωz Xϕ∗ F (z) = Ωz Xϕ∗ F (z) dt t=0 t dt t=0 t d = d(ϕ∗ F )(z) t dt t=0 = (dXH [F ])(z) = Ωz (XXH [F ] (z)). Thus, d Xϕ∗ F = XXH [F ] . dt t=0 t Therefore, d d {ϕ∗ F, ϕ∗ G} = t t Ωz (Xϕ∗ F (z), Xϕ∗ G (z)) dt t=0 dt t=0 t t = Ωz (XXH [F ] , XG (z)) + Ωz (XF (z), XXH [G] (z)) = {XH [F ], G}(z) + {F, XH [G]}(z). Lie Algebras and Jacobi’s Identity. The above development leads to important insight into Poisson brackets. Proposition 5.5.4. The functions F(P ) form a Lie algebra under the Poisson bracket. Proof. Since {F, G} is obviously real bilinear and skew-symmetric, the only thing to check is Jacobi’s identity. From {F, G} = iXF Ω(XG ) = dF (XG ) = XG [F ], we have {{F, G}, H} = XH [{F, G}] and so by Corollary 5.5.3 we get {{F, G}, H} = {XH [F ], G} + {F, XH [G]} = {{F, H}, G} + {F, {G, H}}, (5.5.4) which is Jacobi’s identity. This derivation gives us additional insight: Jacobi’s identity is just the infinitesimal statement of ϕt being canonical . ........................... 15 July 1998—18h02 ...........................

172. 158 5. Hamiltonian Systems on Symplectic Manifolds In the same spirit, one can check that if Ω is a nondegenerate two-form with the Poisson bracket defined by (5.5.1), then the Poisson bracket satisfies the Jacobi identity if and only if Ω is closed (see Exercise 5.5-1). The Poisson bracket-Lie derivative identity {F, G} = XG [F ] = −XF [G] (5.5.5) we derived in this proof will be useful. Proposition 5.5.5. The set of Hamiltonian vector fields XHam (P ) is a Lie subalgebra of X(P ) and, in fact, [XF , XG ] = −X{F,G} . (5.5.6) Proof. As derivations, [XF , XG ][H] = XF XG [H] − XG XF [H] = XF [{H, G}] − XG [{H, F }] = {{H, G}, F } − {{H, F }, G} = −{H, {F, G}} = −X{F,G} [H], by Jacobi’s identity. Proposition 5.5.6. We have d (F ◦ ϕt ) = {F ◦ ϕt , H} = {F, H} ◦ ϕt , (5.5.7) dt where ϕt is the flow of XH and F ∈ F(P ). Proof. By (5.5.5) and the chain rule, d (F ◦ ϕt )(z) = dF (ϕt (z)) · XH (ϕt (z)) = {F, H}(ϕt (z)). dt Since ϕt is symplectic, this becomes {F ◦ ϕt , H ◦ ϕt }(z) which also equals {F ◦ ϕt , H}(z) by conservation of energy. This proves (5.5.7). Equations in Poisson Bracket Form. Equation (5.5.7), often written more compactly as ˙ F = {F, H}, (5.5.8) is called the equation of motion in Poisson bracket form. We indicated in Chapter 1 why the formulation (5.5.8) is important. ........................... 15 July 1998—18h02 ...........................

173. 5.5 Poisson Brackets on Symplectic Manifolds 159 Corollary 5.5.7. F ∈ F(P ) is a constant of the motion for XH if and only if {F, H} = 0. Proposition 5.5.8. Assume that the functions f, g, and {f, g} are integrable relative to the Liouville volume Λ ∈ Ω2n (P ) on a 2n-dimensional symplectic manifold (P, Ω). Then {f, g}Λ = f iXg Λ = − giXf Λ. P ∂P ∂P Proof. Since £Xg Ω = 0, it follows that £Xg Λ = 0 so that div(f Xg ) = Xg [f ] = {f, g}. Therefore, by Stokes’ theorem {f, g}Λ = div(f Xg )Λ = £f Xg Λ = dif Xg Λ = f iXg Λ, P P P P ∂P the second equality following by skew-symmetry of the Poisson bracket. Corollary 5.5.9. Assume that f, g, h ∈ F(P ) have compact support or decay fast enough such that they and their Poisson brackets are L2 integrable relative to the Liouville volume on a 2n-dimensional symplectic manifold (P, Ω). Assume also that at least one of f and g vanish on ∂P , if ∂P = ∅. Then the L2 -inner product is bi-invariant on the Lie algebra (F(P ), { , }), that is, f {g, h}Λ = {f, g}hΛ. P P Proof. From {hf, g} = h{f, g} + f {h, g} we get 0= {hf, g}Λ = h{f, g}Λ + f {h, g}Λ. P P P However, from Proposition 5.5.8, the integral of {hf, g} over P vanishes since one of f or g vanishes on ∂P . The corollary then follows. Exercises 5.5-1. Let Ω be a nondegenerate two-form on a manifold P . Form Hamil- tonian vector fields and the Poisson bracket using the same definitions as in the symplectic case. Show that Jacobi’s identity holds if and only if the two-form Ω is closed. 5.5-2. Let P be a compact boundaryless symplectic manifold. Show that the space of functions F0 (P ) = {f ∈ F(P ) | P f Λ = 0} is a Lie subalgebra of (F(P ), { , }) isomorphic to the Lie algebra of Hamiltonian vector fields on P . ........................... 15 July 1998—18h02 ...........................

174. 160 5. Hamiltonian Systems on Symplectic Manifolds 5.5-3. Using the complex notation z j = q j +ipj , show that the symplectic form on Cn may be written as n i Ω= dz k ∧ d¯k , z 2 k=1 and the Poisson bracket may be written n 2 ∂F ∂G ∂G ∂F {F, G} = − k k . i ∂z k ∂ z k ¯ ∂z ∂ z ¯ k=1 5.5-4. Let J : C2 → R be deﬁned by 1 J= (|z1 |2 − |z2 |2 ). 2 Show that {H, J} = 0, where H is given in Exercise 5.4-3. 5.5-5. Let (P, Ω) be a 2n-dimensional symplectic manifold. Show that the Poisson bracket may be deﬁned by {F, G}Ωn = γdF ∧ dG ∧ Ωn−1 for a suitable constant γ. 5.5-6. Let ϕ : S −→ P be a coisotropic immersion (see Exercise 5.4-4). Let F, H : P −→ R be smooth functions such that d(ϕ∗ F )(s), (ϕ∗ H)(s) vanish on (Ts φ)−1 ([Ts ϕ(Ts S)]Ω(ϕ(s)) ) for all s ∈ S. Show that ϕ∗ {F, H} depends only on ϕ∗ F and ϕ∗ H. ........................... 15 July 1998—18h02 ...........................

175. This is page 161 Printer: Opaque this 6 Cotangent Bundles In many mechanics problems, the phase space is the cotangent bundle T ∗ Q of a configuration space Q. There is an “intrinsic” symplectic structure on T ∗ Q that can be described in various equivalent ways. Assume first that Q is n-dimensional, and pick local coordinates (q 1 , . . . , q n ) on Q. Since ∗ ∗ (dq 1 , . . . , dq n ) is a basis of Tq Q, we can write any α ∈ Tq Q as α = pi dq i . Then (q , . . . , q , p1 , . . . , pn ) are induced local coordinates on T ∗ Q. Define 1 n the canonical symplectic form on T ∗ Q by Ω = dq i ∧ dpi . This defines a closed two-form Ω that can be checked to be independent of the choice of coordinates (q 1 , . . . , q n ). Observe that Ω is locally constant, that is, it does not explicitly depend on the coordinates (q 1 , . . . , q n , p1 , . . . , pn )Reword of phase space points. In this section we show how to do this construction phrase. intrinsically and we will study this canonical symplectic structure in some detail. 6.1 The Linear Case To motivate a coordinate independent definition of Ω, consider the case in which Q is a vector space W (which could be infinite dimensional), so that T ∗ Q = W × W ∗ . We have already described the canonical two-form on W × W ∗: Ω(w,α) ((u, β), (v, γ)) = γ, u − β, v , (6.1.1)

176. 162 6. Cotangent Bundles where (w, α) ∈ W × W ∗ is the base point, u, v ∈ W , and β, γ ∈ W ∗ . This canonical two-form will be constructed from the canonical one-form Θ, defined as follows: Θ(w,α) (u, β) = α, u . (6.1.2) The next proposition shows that the canonical two-form (6.1.1) is exact: Ω = −dΘ. (6.1.3) We begin with a computation that reconciles these formulas with their coordinate expressions. Proposition 6.1.1. In the finite-dimensional case the symplectic form Ω defined by (6.1.1) can be written Ω = dq i ∧ dpi in coordinates q 1 , . . . , q n on W and corresponding dual coordinates p1 , . . . , pn on W ∗ . The associated canonical one-form is given by Θ = pi dq i and (6.1.3) holds. Proof. If (q 1 , . . . , q n , p1 , . . . , pn ) are coordinates on T ∗ W then ∂ ∂ ∂ ∂ ,... , n, ,... , ∂q 1 ∂q ∂p1 ∂pn denotes the induced basis for T(w,α) (T ∗ W ), and (dq 1 , . . . , dq n , dp1 , . . . , dpn ) denotes the associated dual basis of T(w,α) (T ∗ W ). Write ∗ ∂ ∂ (u, β) = uj j , βj ∂q ∂pj and similarly for (v, γ). Hence (dq i ∧ dpi )(w,α) ((u, β), (v, γ)) = (dq i ⊗ dpi − dpi ⊗ dq i )((u, β), (v, γ)) = dq i (u, β)dpi (v, γ) − dpi (u, β)dq i (v, γ) = ui γi − βi v i . Also, Ω(w,α) ((u, β), (v, γ)) = γ(u) − β(v) = γi ui − βi v i . Thus, Ω = dq i ∧ dpi . Similarly, (pi dq i )(w,α) (u, β) = αi dq i (u, β) = αi ui , and Θ(w,α) (u, β) = α(u) = αi ui . Comparing, we get Θ = pi dq i . Therefore, −dΘ = −d(pi dq i ) = dq i ∧ dpi = Ω. ........................... 15 July 1998—18h02 ...........................

177. 6.2 The Nonlinear Case 163 To verify (6.1.3) for the infinite-dimensional case, use (6.1.2) and the second identity in item 6 of the table at the end of §4.4 to give dΘ(w,α) ((u1 , β1 ), (u2 , β2 )) = [DΘ(w,α) · (u1 , β1 )] · (u2 , β2 ) − [DΘ(w,α) · (u2 , β2 )] · (u1 , β1 ) = β1 , u2 − β2 , u1 , since DΘ(w,α) · (u, β) = β, · . But this equals −Ω(w,α) ((u1 , β1 ), (u2 , β2 )). To give an intrinsic interpretation to Θ, let us prove that Θ(w,α) · (u, β) = α, T(w,α) πW (u, β) , (6.1.4) where πW : W × W ∗ → W is the projection. Indeed (6.1.4) coincides with (6.1.2) since T(w,α) πW : W × W ∗ → W is the projection on the first factor. Exercises 6.1-1 (Jacobi–Haretu Coordinates). Consider the configuration space Q = R3 × R3 × R3 with elements denoted r1 , r2 , and r3 . Call the conjugate momenta p1 , p2 , p3 and equip the phase space T ∗ Q with the canonical symplectic structure Ω. Let j = p1 + p2 + p3 . Let r = r2 − r1 and let s = r3 − 1 (r1 + r2 ). Show that the form Ω pulled back to the level sets 2 of j has the form Ω = dr ∧ dπ + ds ∧ dσ, where the variables π and σ are defined by π = 1 (p2 − p1 ) and σ = p3 . 2 6.2 The Nonlinear Case Definition 6.2.1. Let Q be a manifold. We define Ω = −dΘ, where Θ is the one-form on T ∗ Q defined analogous to (6.1.4), namely Θβ (v) = β, T πQ · v , (6.2.1) where β ∈ T ∗ Q, v ∈ Tβ (T ∗ Q), πQ : T ∗ Q → Q is the projection, and T πQ : T (T ∗ Q) → T Q is the tangent map of πQ . The computations in Proposition 6.1.1 show that (T ∗ Q, Ω = −dΘ) is a symplectic manifold; indeed, in local coordinates with (w, α) ∈ U × W ∗ , where U is open in W , and where (u, β), (v, γ) ∈ W × W ∗ , the two-form Ω = −dΘ is given by Ω(w,α) ((u, β), (v, γ)) = γ(u) − β(v). (6.2.2) Darboux’s theorem and its corollary can be interpreted as asserting that any (strong) symplectic manifold locally looks like W ×W ∗ in suitable local coordinates. ........................... 15 July 1998—18h02 ...........................

178. 164 6. Cotangent Bundles Hamiltonian Vector Fields. For a function H : T ∗ Q → R, the Hamil- tonian vector field XH on the cotangent bundle T ∗ Q is given in canonical cotangent bundle charts U × W ∗ , where U is open in W , by δH δH XH (w, α) = ,− . (6.2.3) δα δw Indeed, denoting XH (w, α) = (w, α, v, γ), for any (u, β) ∈ W × W ∗ we have dH(w,α) · (u, β) = Dw H(w,α) · u + Dα H(w,α) · β δH δH = , u + β, (6.2.4) δw δα which, by definition and (6.2.2), equals Ω(w,α) (XH (w, α), (u, β)) = β, v − γ, u . (6.2.5) Comparing (6.2.4) and (6.2.5) gives (6.2.3). In finite dimensions, (6.2.3) is the familiar right-hand side of Hamilton’s equations. Poisson Brackets. Formula (6.2.3) and the definition of the Poisson bracket show that in canonical cotangent bundle charts, δf δg δg δf {f, g}(w, α) = , − , , (6.2.6) δw δα δw δα which in finite dimensions becomes n ∂f ∂g ∂f ∂g {f, g}(q i , pi ) = − . (6.2.7) i=1 ∂q i ∂pi ∂pi ∂q i Pull Back Characterization. Another characterization of the canonical one-form that is sometimes useful is the following: Proposition 6.2.2. Θ is the unique one-form on T ∗ Q such that α∗ Θ = α (6.2.8) for any local one-form α on Q, where, on the left-hand side, α is regarded as a map (of some open subset of ) Q to T ∗ Q. Proof. In finite dimensions, if α = αi (q j ) dq i and Θ = pi dq i , then to calculate α∗ Θ means that we substitute pi = αi (q j ) into Θ, a process which clearly gives back α, so α∗ Θ = α. The general argument is as follows. If Θ is the canonical one-form on T ∗ Q, and v ∈ Tq Q, then (α∗ Θ)q · v = Θα(q) · Tq α(v) = α(q), Tα(q) πQ (Tq α(v)) = α(q), Tq (πQ ◦ α)(v) = α(q) · v ........................... 15 July 1998—18h02 ...........................

179. 6.2 The Nonlinear Case 165 since πQ ◦ α = identity on Q. For the converse, assume that Θ is a one-form on T ∗ Q satisfying (6.2.8). We will show that it must then be the canonical one-form (6.2.1). In finite dimensions this is straightforward: if Θ = Ai dq i + B i dpi for Ai , B i functions of (q j , pj ), then ∂αi α∗ Θ = (Ai ◦ α) dq i + (B i ◦ α) dαi = Aj ◦ α + (B i ◦ α) dq j ∂q j which equals α = αi dq i if and only if ∂αi Aj ◦ α + (B i ◦ α) = αj . ∂q j Since this must hold for all αj , putting α1 , . . . , αn constant it follows that Aj ◦ α = αj , that is, Aj = pj . Therefore, the remaining equation is ∂αi (B i ◦ α) =0 ∂q j for any αi ; choosing αi (q 1 , . . . , q n ) = q0 + (q i − q0 )p0 (no sum) implies i i i j 0 0 = (B j ◦ α)(q0 , . . . , q0 )p0 for all (q0 , pj ); therefore, B j = 0 and thus 1 n j Θ = pi dq i .1 Exercises 6.2-1. Let N be a submanifold of M and denote by ΘN and ΘM the canonical one-forms on the cotangent bundles πN : T ∗ N → N and πM : T ∗ M → M , respectively. Let π : (T ∗ M )|N → T ∗ N be the projection ∗ defined by π(αn ) = αn |Tn N , where n ∈ N and αn ∈ Tn M . Show that ∗ ∗ ∗ ∗ π ΘN = i ΘM , where i : (T M )|N → T M is the inclusion. 6.2-2. Let f : Q → R and X ∈ X(T ∗ Q). Show that Θ(X) ◦ df = X[f ◦ πQ ] ◦ df. 1 In infinite dimensions, the proof is slightly different. We will show that if (6.2.8) holds then Θ is locally given by (6.1.4) and thus it is the canonical one-form. If U ⊂ E is the chart domain the Banach space E modeling Q for any v ∈ E we have (α∗ Θ)u · (u, v) = Θ(u, α(u)) · (v, Dα(u) · v), where α is given locally by u → (u, α(u)) for α : U → E ∗ . Thus (6.2.8) is equivalent to Θ(u,α(u)) · (v, Dα(u) · v) = α(u), v which would imply (6.1.4) and hence Θ being the canonical one-form, provided we can show that for prescribed γ, δ ∈ E ∗ , u ∈ U, v ∈ E there is an α : U → E ∗ such that α(u) = γ, Dα(u) · v = δ. Such a mapping is constructed in the following way. For v = 0 choose α(u) to equal γ for all u. For v = 0, by the Hahn-Banach theorem one can find a ϕ ∈ E ∗ such that ϕ(v) = 1. Now set α(x) = γ − ϕ(u)δ + ϕ(x)δ.] ........................... 15 July 1998—18h02 ...........................

180. 166 6. Cotangent Bundles 6.2-3. Let Q be a given configuration manifold and let the extended phase space be defined by (T ∗ Q) × R. Given a time dependent vector field X on T ∗ Q, extend it to a vector field X on (T ∗ Q) × R by X = (X, 1). ˜ ˜ ∗ Let H be a (possibly time-dependent) function on (T Q) × R and set ΩH = Ω + dH ∧ dt, where Ω is the canonical two-form. Show that X is the Hamiltonian vector field for H if and only if iX ΩH = 0. ˜ 6.2-4. Give an example of a symplectic manifold (P, Ω), where Ω is exact, but P is not a cotangent bundle. 6.3 Cotangent Lifts We now describe an important way to create symplectic transformations on cotangent bundles. Definition 6.3.1. Given two manifolds Q and S and a diffeomorphism f : Q → S, the cotangent lift T ∗ f : T ∗ S → T ∗ Q of f is defined by T ∗ f (αs ), v = αs , (T f · v) , (6.3.1) where ∗ αs ∈ Ts S, v ∈ Tq Q, and s = f (q). The importance of this construction is that T ∗ f is guaranteed to be symplectic; it is often called a “point transformation” because it arises from a diffeomorphism on points in configuration space. Notice that while T f covers f , T ∗ f covers f −1 . Denote by πQ : T ∗ Q → Q and πS : T ∗ S → S, the canonical cotangent bundle projections. Proposition 6.3.2. A diffeomorphism ϕ : T ∗ S → T ∗ Q preserves the canonical one-forms ΘQ and ΘS on T ∗ Q and T ∗ S, respectively, if and only if ϕ is the cotangent lift T ∗ f of some diffeomorphism f : Q → S. Proof. First assume that f : Q → S is a diffeomorphism. Then for arbitrary β ∈ T ∗ S and v ∈ Tβ (T ∗ S), we have ((T ∗ f )∗ ΘQ )β · v = (ΘQ )T ∗ f (β) · T T ∗ f (v) = T ∗ f (β), (T πQ ◦ T T ∗ f ) · v = β, T (f ◦ πQ ◦ T ∗ f ) · v = β, T πS · v = ΘSβ · v, ........................... 15 July 1998—18h02 ...........................

181. 6.3 Cotangent Lifts 167 since f ◦ πQ ◦ T ∗ f = πS . Conversely, assume that ϕ∗ ΘQ = ΘS , that is, ϕ(β), T (πQ ◦ ϕ)(v) = β, T πS (v) (6.3.2) for all β ∈ T ∗ S and v ∈ Tβ (T ∗ S). Since ϕ is a diffeomorphism, the range of Tβ (πQ ◦ ϕ) is TπQ (ϕ(β)) Q, so that letting β = 0 in (6.3.2) implies that ϕ(0) = 0. Arguing similarly for ϕ−1 instead of ϕ, we conclude that ϕ restricted to the zero section S of T ∗ S is a diffeomorphism onto the zero section Q of T ∗ Q. Define f : Q → S by f = ϕ−1 |Q. We will show below that ϕ is fiber-preserving or, equivalently, that f ◦ πQ = πS ◦ ϕ−1 . For this we need the following: Lemma 6.3.3. Define the flow FtQ on T ∗ Q by FtQ (α) = et α and let VQ be the vector field it generates. Then ΘQ , VQ = 0, £VQ ΘQ = ΘQ , and iVQ ΩQ = −ΘQ . (6.3.3) Proof. Since FtQ is fiber-preserving, VQ will be tangent to the fibers and hence T πQ ◦ VQ = 0. This implies by (6.2.1) that ΘQ , VQ = 0. To prove the second formula, note that πQ ◦ FtQ = πQ . Let α ∈ Tq Q, v ∈ Tα (T ∗ Q), ∗ and Θα denote ΘQ evaluated at α. We have ((FtQ )∗ Θ)α · v = ΘF Q (α) · T FtQ (v) t = FtQ (α), (T πQ ◦ T FtQ )(v) = et α, T (πQ ◦ FtQ )(v) = et α, T πQ (v) = et Θα · v, that is, (FtQ )∗ ΘQ = et ΘQ . Taking the derivative relative to t at t = 0 yields the second formula. Finally, the first two formulas imply iVQ ΩQ = −iVQ dΘQ = −£VQ ΘQ + diVQ ΘQ = −ΘQ . Continuing the proof of the proposition, note that by (6.3.3) we have iϕ∗ VQ ΩS = iϕ∗ VQ ϕ∗ ΩQ = ϕ∗ (iVQ ΩQ ) = −ϕ∗ ΘQ = −ΘS = iVS ΩS , so that weak nondegeneracy of ΩS implies ϕ∗ VQ = VS . Thus ϕ commutes with the flows FtQ and FtS , that is, for any β ∈ T ∗ S we have ϕ(et β) = ........................... 15 July 1998—18h02 ...........................

182. 168 6. Cotangent Bundles et ϕ(β). Letting t → −∞ in this equality implies (ϕ ◦ πS )(β) = (πQ ◦ ϕ)(β) since et β → πS (β) and et ϕ(β) → (πQ ◦ ϕ)(β) for t → −∞. Thus πQ ◦ ϕ = ϕ ◦ πS , or f ◦ πQ = πS ◦ ϕ−1 . Finally, we show that T ∗ f = ϕ. For β ∈ T ∗ S, v ∈ Tβ (T ∗ S), (6.3.2) gives T ∗ f (β), T (πQ ◦ ϕ)(v) = β, T (f ◦ πQ ◦ ϕ)(v) = β, T πS (v) = (ΘS )β · v = (ϕ∗ ΘQ )β · v = (ΘQ )ϕ(β) · Tβ ϕ(v) = ϕ(β), Tβ (πQ ◦ ϕ)(v) , which shows that T ∗ f = ϕ since the range of Tβ (πQ ◦ ϕ) is the whole tangent space at (πQ ◦ ϕ)(β) to Q. In finite dimensions, the first part of this proposition can be seen in coordinates as follows. Write (s1 , . . . , sn ) = f (q 1 , . . . , q n ) and define ∂si pj = ri , (6.3.4) ∂q j where (q 1 , . . . , q n , p1 , . . . , pn ) are cotangent bundle coodinates on T ∗ Q and (s1 , . . . , sn , r1 , . . . , rn ) on T ∗ S. Since f is a diffeomorphism, it determines the q i in terms of the sj , say q i = q i (s1 , . . . , sn ), so both q i and pj are functions of (s1 , . . . , sn , r1 , . . . , rn ). The map T ∗ f is given by (s1 , . . . , sn , r1 , . . . , rn ) → (q 1 , . . . , q n , p1 , . . . , pn ). (6.3.5) To see that (6.3.5) preserves the canonical one-form, use the chain rule and (6.3.4): ∂si k ri dsi = ri dq = pk dq k . (6.3.6) ∂q k Note that if f and g are diffeomorphisms of Q, then T ∗ (f ◦ g) = T ∗ g ◦ T ∗ f, (6.3.7) that is, the cotangent lift switches the order of compositions; in fact, it is useful to think of T ∗ f as the adjoint of T f . Exercises 6.3-1. The Lorentz group L is the group of invertible linear transformations of R4 to itself that preserve the quadratic form x2 + y 2 + z 2 − c2 t2 , where c is a constant, the speed of light. Describe all elements of this group. Let Λ0 denote one of these transformations. Map L to itself by Λ → Λ0 Λ. Calculate the cotangent lift of this map. 6.3-2. We have shown that a transformation of T ∗ Q is the cotangent lift of a diffeomorphism of configuration space if and only if it preserves the canonical one-form. Find this result in Whittaker’s book. ........................... 15 July 1998—18h02 ...........................

183. 6.4 Lifts of Actions 169 6.4 Lifts of Actions A left action of a group G on a manifold M associates to each group element g ∈ G a diffeomorphism Φg of M , such that Φgh = Φg ◦ Φh . Thus, the collection of Φg ’s is a group of transformations of M . If we replace the condition Φgh = Φg ◦ Φh by Ψgh = Ψh ◦ Ψg we speak of a right action. We often write Φg (m) = g · m and Ψg (m) = m · g for m ∈ M . Definition 6.4.1. Let Φ be an action of a group G on a manifold Q. The right lift Φ∗ of the action Φ to the symplectic manifold T ∗ Q is the right action defined by the rule Φ∗ (α) = (Tg−1 ·q Φg )(α), g ∗ (6.4.1) where g ∈ G, α ∈ Tq Q, and T ∗ Φg is the cotangent lift of the diffeomorphism ∗ Φg : Q → Q. By (6.3.7), we see that Φ∗ = T ∗ Φgh = T ∗ (Φg ◦ Φh ) = T ∗ Φh ◦ T ∗ Φg = Φ∗ ◦ Φ∗ gh h g (6.4.2) so Φ∗ is a right action. To get a left action, denoted Φ∗ and called the left lift of Φ, one sets ∗ (Φ∗ )g = Tg·q (Φg−1 ). (6.4.3) In either case these lifted actions are actions by canonical transformations because of Proposition 6.3.2. We shall return to the study of actions of groups after we study Lie groups in Chapter 9. Examples (a) For a system of N particles in R3 , we choose the configuration space Q = R3N . We write (qj ) for an N -tuple of vectors labeled by j = 1, . . . , N . Similarly, elements of the momentum phase space P = T ∗ R3N ∼ R6N ∼= = R3N × R3N are denoted (qj , pj ). Let the additive group G = R3 of translations act on Q according to Φx (qj ) = qj + x, where x ∈ R3 . (6.4.4) Each of the N position vectors qj is translated by the same vector x. Lifting the diffeomorphism Φx : Q → Q, we obtain an action Φ∗ of G on P . We assert that Φ∗ (qj , pj ) = (qj − x, pj ). x (6.4.5) To verify (6.4.5), observe that T Φx : T Q → T Q is given by (qi , qj ) → (qi + x, qj ) ˙ ˙ (6.4.6) so its dual is (qi , pj ) → (qi − x, pj ). ........................... 15 July 1998—18h02 ...........................

184. 170 6. Cotangent Bundles (b) Consider the action of GL(n, R), the group of n × n invertible matrices, or more properly, the group of invertible linear transformations of Rn to itself, on Rn given by ΦA (q) = Aq. (6.4.7) The group of induced canonical transformations of T ∗ Rn to itself is given by Φ∗ (q, p) = (A−1 q, AT p), A (6.4.8) which is readily verified. Notice that this reduces to the same transformation of q and p when A is orthogonal. Exercises 6.4-1. Let the multiplicative group R {0} act on Rn by Φλ (q) = λq. Calculate the cotangent lift of this action. 6.5 Generating Functions Consider a symplectic diffeomorphism ϕ : T ∗ Q1 → T ∗ Q2 described by functions pi = pi (q j , sj ), ri = ri (q j , sj ), (6.5.1) where (q i , pi ) and (sj , rj ) are cotangent coordinates on T ∗ Q1 and on T ∗ Q2 , respectively. In other words, assume that we have a map Γ : Q1 × Q2 → T ∗ Q1 × T ∗ Q2 (6.5.2) whose image is the graph of ϕ. Let Θ1 be the canonical one-form on T ∗ Q1 and Θ2 be that on T ∗ Q2 . By definition, d(Θ1 − ϕ∗ Θ2 ) = 0. (6.5.3) This implies, in view of (6.5.1), that pi dq i − ri dsi (6.5.4) is closed. Restated, Γ∗ (Θ1 − Θ2 ) is closed. This holds if (and implies locally by the Poincar´ lemma) e Γ∗ (Θ1 − Θ2 ) = dS (6.5.5) ........................... 15 July 1998—18h02 ...........................

185. 6.5 Generating Functions 171 for a function S(q, s). In coordinates, (6.5.5) reads ∂S i ∂S i pi dq i − ri dsi = dq + i ds (6.5.6) ∂q i ∂s which is equivalent to ∂S ∂S pi = , ri = − . (6.5.7) ∂q i ∂si One calls S a generating function for the canonical transformation. With generating functions of this sort, one may run into singularities even with the identity map! See Exercise6.5-1. Presupposed relations other than (6.5.1) lead to different conclusions than (6.5.7). Point transformations are generated in this sense; if S(q i , rj ) = sj (q)rj , then ∂S ∂S si = and pi = . (6.5.8) ∂ri ∂q i (Here one writes pi dq i + si dri = dS.) In general, consider a diffeomorphism ϕ : P1 → P2 of one symplectic manifold (P1 , Ω1 ) to another (P2 , Ω2 ) and denote the graph of ϕ, by Γ(ϕ) ⊂ P1 × P2 . ∗ ∗ Let iϕ : Γ(ϕ) → P1 × P2 be the inclusion and let Ω = π1 Ω1 − π2 Ω2 , where πi : P1 × P2 → Pi is the projection. One verifies that ϕ is symplectic if and only if i∗ Ω = 0. Indeed, since π1 ◦ iϕ is the projection restricted to Γ(ϕ) ϕ and π2 ◦ iϕ = ϕ ◦ π1 on Γ(ϕ), it follows that i∗ Ω = (π1 |Γ(ϕ))∗ (Ω1 − ϕ∗ Ω2 ), ϕ and hence i∗ Ω = 0 if and only if ϕ is symplectic because (π1 |Γ(ϕ))∗ is in- ϕ jective. In this case, one says Γ(ϕ) is an isotropic submanifold of P1 × P2 (equipped with the symplectic form Ω); in fact, since Γ(ϕ) has half the dimension of P1 × P2 , it is maximally isotropic, or a Lagrangian manifold . Now suppose one chooses a form Θ such that Ω = −dΘ. Then i∗ Ω = ϕ −di∗ Θ = 0, so locally on Γ(ϕ) there is a function S : Γ(ϕ) → R such that ϕ i∗ Θ = dS. ϕ (6.5.9) This defines the generating function of the canonical transformation ϕ. Since Γ(ϕ) is diffeomorphic to P1 and also to P2 we can regard S as a function on P1 or P2 . If P1 = T ∗ Q1 and P2 = T ∗ Q2 , we can equally well regard (at least locally) S as defined on Q1 × Q2 . In this way, the general construction of generating functions reduces to the case in equations (6.5.7) and (6.5.8) above. By making other choices of Q, the reader can construct ........................... 15 July 1998—18h02 ...........................

186. 172 6. Cotangent Bundles other generating functions and reproduce formulas in, for instance, Gold- stein [1980] or Whittaker [1927]. The approach here is based on Sniatycki and Tulczyjew [1971]. Generating functions play an important role in Hamilton-Jacobi theory, in the quantum-classical mechanical relationship (where S plays the role of the quantum mechanical phase), and in numerical integration schemes for Hamiltonian systems. We shall see a few of these aspects later on. Exercises 6.5-1. Show that 1 S(q i , sj , t) = q−s 2 2t generates a canonical transformation that is the identity at t = 0. 6.5-2. (A first-order symplectic integrator). Given H, let S(q i , rj , t) = rk q k − tH(q i , rj ). Show that S generates a canonical transformation which is a first-order approximation to the flow of XH for small t. 6.6 Fiber Translations and Magnetic Terms Momentum Shifts. We saw above that cotangent lifts provide a basic construction of canonical transformations. Fiber translations provide a second. Proposition 6.6.1 (Momentum Shifting Lemma). Let A be a one- form on Q and let tA : T ∗ Q → T ∗ Q be defined by αq → αq + A(q), where αq ∈ Tq Q. Let Θ be the canonical one-form on T ∗ Q. Then ∗ t∗ Θ = Θ + πQ A, A ∗ (6.6.1) where πQ : T ∗ Q → Q is the projection. Hence t∗ Ω = Ω − πQ dA, A ∗ (6.6.2) where Ω = −dΘ is the canonical symplectic form. Thus, tA is a canonical transformation if and only if dA = 0. Proof. We prove this using a finite-dimensional coordinate computation. The reader is asked to supply the coordinate-free and infinite-dimensional proofs as an exercise. In coordinates, tA is the map tA (q i , pj ) = (q i , pj + Aj ). (6.6.3) ........................... 15 July 1998—18h02 ...........................

187. 6.6 Fiber Translations and Magnetic Terms 173 Thus, t∗ Θ = t∗ (pi dq i ) = (pi + Ai )dq i = pi dq i + Ai dq i , A A (6.6.4) ∗ which is the coordinate expression for Θ + πQ A. The remaining assertions follow directly from this. In particular, fiber translation by the differential of a function A = df is a canonical transformation; in fact, f induces, in the sense of the preceding section, a generating function (see Exercise 6.6-2). The two basic classes of canonical transformations, lifts, and fiber translations, play an important part in mechanics. Magnetic Terms. A symplectic form on T ∗ Q, different from the canonical one, is obtained in the following way. Let B be a closed two-form on Q. Then Ω − πQ B is a closed two-form on T ∗ Q, where Ω is the canonical ∗ ∗ two-form. To see that Ω−πQ B is (weakly) nondegenerate, use the fact that in a local chart this form is given at the point (w, α) by ((u, β), (v, γ)) → γ, u − β, v − B(w)(u, v). (6.6.5) Proposition 6.6.2. (i) Let Ω be the canonical two-form on T ∗ Q and let πQ : T ∗ Q → Q be the projection. If B is a closed two-form on Q, then ∗ ΩB = Ω − πQ B (6.6.6) is a (weak ) symplectic form on T ∗ Q. (ii) Let B and B be closed two-forms on Q and assume that B − B = dA. Then the mapping tA (fiber translation by A) is a symplectic diffeomorphism of (T ∗ Q, ΩB ) with (T ∗ Q, ΩB ). Proof. Part (i) follows by an argument similar to that in the momentum shifting lemma. For (ii), use formula (6.6.2) to get t∗ Ω = Ω − πQ dA = Ω − πQ B + πQ B , A ∗ ∗ ∗ (6.6.7) so that t∗ (Ω − πQ B ) = Ω − πQ B A ∗ ∗ since πQ ◦ tA = πQ . Symplectic forms of the type ΩB arise in the reduction process.2 In the ∗ following section, we explain why the extra term πQ B is called a magnetic term. 2 Magnetic terms come up in what is called the cotangent bundle reduction theorem; see Smale [1972], Abraham and Marsden [1978], Kummer [1981], Nill [1983], Montgomery, Marsden, and Ratiu [1984], Gozzi and Thacker [1987], and Marsden [1992]. ........................... 15 July 1998—18h02 ...........................

188. 174 6. Cotangent Bundles Exercises 6.6-1. Provide the intrinsic proof of Proposition 6.6.1. 6.6-2. If A = df , use a coordinate calculation to check that S(q i , ri ) = ri q i − f (q i ) is a generating function for tA . 6.7 A Particle in a Magnetic Field Let B be a closed two-form on R3 and let B = Bx i + By j + Bz k be the associated divergence-free vector ﬁeld, that is, iB (dx ∧ dy ∧ dz) = B, so that B = Bx dy ∧ dz − By dx ∧ dz + Bz dx ∧ dy. Thinking of B as a magnetic ﬁeld, the equations of motion for a particle with charge e and mass m are given by the Lorentz force law : dv e m = v × B, (6.7.1) dt c where v = (x, y, z). On R3 × R3 , that is, (x, v)-space, consider the sym- ˙ ˙ ˙ plectic form e ΩB = m(dx ∧ dx + dy ∧ dy + dz ∧ dz) − B ˙ ˙ ˙ (6.7.2) c that is, (6.6.6). As Hamiltonian, take the kinetic energy: m H = (x2 + y 2 + z 2 ) ˙ ˙ ˙ (6.7.3) 2 writing XH (u, v, w) = (u, v, w, u, v, w), the condition ˙ ˙ ˙ dH = iXH ΩB (6.7.4) is m(x dx + y dy + z dz) = m(u dx − u dx + v dy − v dy ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ e + w dz − w dz) − [Bx v dz − Bx w dy ˙ ˙ c − By u dz + By w dx + Bz u dy − Bz v dx], which is equivalent to u = x, ˙ v = y, w = z, ˙ ˙ e mu = (Bz v − By w), ˙ c e mv = (Bx w − Bz u), ˙ c e mw = (By u − Bx v), ˙ c ........................... 15 July 1998—18h02 ...........................

189. 6.7 A Particle in a Magnetic Field 175 that is, to e  m¨ = (Bz y − By z), x ˙ ˙   c   e  m¨ = (Bx z − Bz x), y ˙ ˙ (6.7.5) c   e   m¨ = (By x − Bx y), z ˙ ˙  c which is the same as (6.7.1). Thus the equations of motion for a particle in a magnetic field are Hamiltonian, with energy equal to the kinetic energy and with the symplectic form ΩB . If B = dA; that is, B = ×A, where A = A, then the map tA : (x, v) → (x, p), where p = mv + eA/c pulls back the canonical form to ΩB by the momentum shifting lemma. Thus, equations (6.7.1) are also Hamiltonian relative to the canonical bracket on (x, p)-space with the Hamiltonian 1 e HA = p − A 2. (6.7.6) 2m c Remarks. 1. Not every magnetic field can be written as B = × A on Euclidean space. For example, the field of a magnetic monopole of strength g = 0, namely r B(r) = g , (6.7.7) r 3 cannot be written this way since the flux of B through the unit sphere is 4πg, yet Stokes’ theorem applied to the two-sphere would give zero; see Exercise 4.4-3. Thus, one might think that the Hamiltonian formulation involving only B (that is, using ΩB and H) is preferable. However, there is a way to recover the magnetic potential A by regarding it as a connection on a nontrivial bundle over R3 {0}. (This bundle over the sphere S 2 is the Hopf fibration S 3 → S 2 .) This same construction can be carried out using reduction and we shall do so later. For a readable account of some aspects of this situation, see Yang [1985]. 2. When one studies the motion of a particle in a Yang–Mills field, one finds a beautiful generalization of this construction and related ideas using the theory of principal bundles; see Sternberg [1977], Weinstein [1978], and Montgomery [1984]. 3. In Chapter 8 we study centrifugal and Coriolis forces and will see some structures analogous to those here. ........................... 15 July 1998—18h02 ...........................

190. 176 6. Cotangent Bundles Exercises 6.7-1. Show that particles in constant magnetic ﬁelds move in helixes. 6.7-2. Verify “by hand” that 1 m v 2 2 is conserved for a particle moving in a magnetic ﬁeld. 6.7-3. Verify “by hand” that Hamilton’s equations for HA are the Lorentz force law equations (6.7.1). ........................... 15 July 1998—18h02 ...........................

191. This is page 177 Printer: Opaque this 7 Lagrangian Mechanics Our approach so far has emphasized the Hamiltonian point of view. How- ever, there is an independent point of view, that of Lagrangian mechanics, based on variational principles. This alternative viewpoint, computational convenience, and the fact that the Lagrangian is very useful in covariant relativistic theories, can be used as arguments for the importance of the Lagrangian formulation. Ironically it was Hamilton [1830] who discovered the variational basis of Lagrangian mechanics. 7.1 Hamilton’s Principle of Critical Action Much of mechanics can be based on variational principles. Indeed, it is the variational formulation that is the most covariant, being useful for relativistic systems as well. In the next chapter we shall see the utility of the Lagrangian approach in the study of rotating frames and moving systems in and we will use it as one way to approach Hamilton–Jacobi theory. Consider a conﬁguration manifold Q and the velocity phase space T Q. We consider a function L : T Q → R called the Lagrangian. Speaking informally, Hamilton’s principle of critical action states that dq i δ L qi , dt = 0, (7.1.1) dt where we take variations amongst paths q i (t) in Q with ﬁxed endpoints. (We will study this process a little more carefully in §8.1.) Taking the

192. 178 7. Lagrangian Mechanics variation in (7.1.1), the chain rule gives ∂L i ∂L d i δq + i δq dt (7.1.2) ∂q i ∂ q dt ˙ for the left-hand side. Integrating the second term by parts and using the boundary conditions δq i = 0 at the endpoints of the time interval in question, we get ∂L d ∂L − δq i dt = 0. (7.1.3) ∂q i dt ∂ qi ˙ If this is to hold for all such variations δq i (t), then ∂L d ∂L − = 0, (7.1.4) ∂q i dt ∂ q i ˙ which are the Euler–Lagrange equations. We set pi = ∂L/∂ q i , assume that the transformation (q i , q j ) → (q i , pj ) ˙ ˙ is invertible and we deﬁne the Hamiltonian by H(q i , pj ) = pi q i − L(q i , q i ). ˙ ˙ (7.1.5) Note that ∂H qi = ˙ , ∂pi since ∂H ∂ qj ˙ ∂L ∂ q j ˙ = q i + pj ˙ − j = qi ˙ ∂pi ∂pi ∂ q ∂pi ˙ from (7.1.5) and the chain rule. Likewise, ∂H pi = − ˙ ∂q i from (7.1.4) and ∂H ∂ qi ˙ ∂L ∂L ∂ q i ˙ ∂L j = pi j − j − i j = − j . ∂q ∂q ∂q ∂ q ∂q ˙ ∂q In other words, the Euler-Lagrange equations are equivalent to Hamilton’s equations. Thus, it is reasonable to explore the geometry of the Euler-Lagrange equations using the canonical form on T ∗ Q pulled back to T Q using pi = ∂L/∂ q i . We do this in the next sections. ˙ This is one standard way to approach the geometry of the Euler–Lagrange equations. Another is to use the variational principle itself. The reader will notice that the canonical one-form pi dq i appears as the boundary terms when we take the variations. This can in fact be used as a basis for the introduction of the canonical one-form in Lagrangian mechanics. We shall develop this approach in Chapter 8. See also Exercise 7.2-2. ........................... 15 July 1998—18h02 ...........................

193. 7.2 The Legendre Transform 179 Exercises 7.1-1. Verify that the Euler–Lagrange and Hamilton equations are equivalent, even if L is time-dependent. 7.1-2. Show that the conservation of energy equation results if, in Hamil- ton’s principle, variations corresponding to reparametrizations of the given curve q(t) are chosen. 7.2 The Legendre Transform Fiber Derivatives. Given a Lagrangian L : T Q → R, define a map FL : T Q → T ∗ Q, called the fiber derivative, by d FL(v) · w = L(v + sw), (7.2.1) ds s=0 where v, w ∈ Tq Q. Thus, FL(v) · w is the derivative of L at v along the fiber Tq Q in the direction w. Note that FL is fiber-preserving; that is, it maps ∗ the fiber Tq Q to the fiber Tq Q. In a local chart U × E for T Q, where U is open in the model space E for Q, the fiber derivative is given by FL(u, e) = (u, D2 L(u, e)), (7.2.2) where D2 L denotes the partial derivative of L with respect to its second argument. For finite–dimensional manifolds, with (q i ) denoting coordinates on Q and (q i , q i ) the induced coordinates on T Q, the fiber derivative has ˙ the expression ∂L FL(q i , q i ) = ˙ qi , , (7.2.3) ∂ qi ˙ that is, FL is given by ∂L pi = . (7.2.4) ∂ qi ˙ The associated energy function is defined by E(v) = FL(v) · v − L(v). In many examples it is the relationship (7.2.4) that gives physical meaning to the momentum variables. We call FL the Legendre transform. Lagrangian Forms. Let Ω denote the canonical symplectic form on T ∗ Q. Using FL, we obtain a one-form ΘL and a closed two–form ΩL on T Q by setting ΘL = (FL)∗ Θ and ΩL = (FL)∗ Ω. (7.2.5) ........................... 15 July 1998—18h02 ...........................

194. 180 7. Lagrangian Mechanics We call ΘL the Lagrangian one-form and ΩL the Lagrangian two– form. Since d commutes with pull back, we get ΩL = −dΘL . Using the local expressions for Θ and Ω, a straightforward pull–back computation yields the following local formula for ΘL and ΩL : if E is the model space for Q, U is the range in E of a chart on Q, and U × E is the corresponding range of the induced chart on T Q, then for (u, e) ∈ U × E and tangent vectors (e1 , e2 ), (f1 , f2 ) in E × E, we have T(u,e) FL · (e1 , e2 ) = (u, D2 L(u, e), e1 , D1 (D2 L(u, e)) · e1 + D2 (D2 L(u, e)) · e2 ), (7.2.6) so that using the local expression for Θ and the definition of pull back, ΘL (u, e) · (e1 , e2 ) = D2 L(u, e) · e1 . (7.2.7) Similarly, one finds that ΩL (u, e) · ((e1 , e2 ), (f1 , f2 )) = D1 (D2 L(u, e) · e1 ) · f1 − D1 (D2 L(u, e) · f1 ) · e1 + D2 D2 L(u, e) · e1 · f2 − D2 D2 L(u, e) · f1 · e2 , (7.2.8) where D1 and D2 denote the first and second partial derivatives. In finite dimensions, formulae (7.2.6) and (7.2.7) or a direct pull–back of pi dq i and dq i ∧ dpi yields ∂L i ΘL = dq (7.2.9) ∂ qi ˙ and ∂2L ∂2L ΩL = i ∂q j dq i ∧ dq j + i j dq i ∧ dq j , ˙ (7.2.10) ∂q ˙ ∂q ∂q ˙ ˙ (a sum on all i, j is understood). As a 2n × 2n skew–symmetric matrix,   ∂2L  A ∂ qi ∂ qj  ˙ ˙ ΩL =   2 ,  (7.2.11) ∂ L − i j 0 ∂q ∂q ˙ ˙ where A is the skew–symmetrization of ∂ 2 L/∂ q i ∂q j . From these expres- ˙ sions, it follows that ΩL is (weakly) nondegenerate if and only if the quadratic form D2 D2 L(u, e) is (weakly) nondegenerate. In this case, we say L is a regular or nondegenerate Lagrangian. The implicit function theorem shows that the fiber derivative is locally invertible if and only if L is regular. ........................... 15 July 1998—18h02 ...........................

195. 7.3 Euler–Lagrange Equations 181 Exercises 7.2-1. Let m 2 2 2 L(q 1 , q 2 , q 3 , q 1 , q 2 , q 3 ) = ˙ ˙ ˙ q1 ˙ + q2 ˙ + q3 ˙ + q1 q1 + q2 q2 + q3 q3 . ˙ ˙ ˙ 2 Calculate ΘL , ΩL and the corresponding Hamiltonian. 7.2-2. For v ∈ Tq Q, define its vertical lift v l ∈ Tv (T Q) to be the tangent vector to the curve v + tv at t = 0. Show that ΘL may be defined by w ΘL = v l dL, where w ∈ Tv (T Q) and where w ΘL = iw ΘL is the interior product. Also, show that the energy is E(v) = v l dL − L(v). 7.2-3 (Abstract Legendre Transform). Let V be a vector bundle over a manifold S and let L : V → R. For v ∈ V , let ∂L w= ∈ v∗ ∂v denote the fiber derivative. Assume that the map v → w is a local diffeomorphism and let H : V ∗ → R be defined by H(w) = w, v − L(v). Show that ∂H v= . ∂w 7.3 Euler–Lagrange Equations Hyperregular Lagrangians. Given a Lagrangian L, the action of L is the map A : T Q → R that is defined by A(v) = FL(v) · v, and, as we defined above, the energy of L is E = A − L. In charts, A(u, e) = D2 L(u, e) · e, (7.3.1) E(u, e) = D2 L(u, e) · e − L(u, e), (7.3.2) and in finite dimensions, (7.3.1) and (7.3.2) read ∂L A(q i , q i ) = q i ˙ ˙ = pi q i , ˙ (7.3.3) ∂ qi ˙ ∂L E(q i , q i ) = q i i − L(q i , q i ) = pi q i − L(q i , q i ). ˙ ˙ ˙ ˙ ˙ (7.3.4) ∂q˙ ........................... 15 July 1998—18h02 ...........................

196. 182 7. Lagrangian Mechanics If L is a Lagrangian such that FL : T Q → T ∗ Q is a diffeomorphism, we say L is a hyperregular Lagrangian. In this case, set H = E ◦ (FL)−1 . Then XH and XE are FL–related since FL is, by construction, symplectic. Thus, hyperregular Lagrangians on T Q induce Hamiltonian systems on T ∗ Q. Conversely, one can show that hyperregular Hamiltonians on T ∗ Q come from Lagrangians on T Q (see §7.4 for definitions and details). Lagrangian Vector Field. More generally, a vector field Z on T Q is called a Lagrangian vector field or a Lagrangian system for L, if the Lagrangian condition ΩL (v)(Z(v), w) = dE(v) · w (7.3.5) holds for all v ∈ Tq Q and w ∈ Tv (T Q). If L is regular, so that, ΩL is a (weak) symplectic form, there would exist at most one such Z, which would be the Hamiltonian vector field of E with respect to the (weak) symplectic form ΩL . In this case we know that E is conserved on the flow of Z. In fact, the same result holds, even if L is degenerate: Proposition 7.3.1. Let Z be a Lagrangian vector field for L and let v(t) ∈ T Q be an integral curve of Z. Then E(v(t)) is constant in t. Proof. By the chain rule, d E(v(t)) = dE(v(t)) · v(t) = dE(v(t)) · Z(v(t)) ˙ dt = ΩL (v(t))(Z(v(t))), Z(v(t)) = 0 (7.3.6) by skew–symmetry of ΩL . We usually assume ΩL is nondegenerate, but the degenerate case comes up in the Dirac theory of constraints (see Dirac [1950, 1964], Kunzle [1969], Hansen, Regge, and Teitelboim [1976], Gotay, Nester, and Hinds [1979] references therein, and §8.5). Second-Order Equations. The vector field Z often has a special property, namely, Z is a second-order equation. Definition 7.3.2. A vector field V on T Q is called a second-order equation provided T τQ ◦V = identity, where τQ : T Q → Q is the canonical projection. If c(t) is an integral curve of V, (τQ ◦ c)(t) is called the base integral curve of c(t). It is easy to see that the condition for V being second-order is equivalent to the following: for any chart U × E on T Q, we can write V (u, e) = ((u, e), (e, V2 (u, e))), for some map V2 : U × E → E. Thus, the dynamics is determined by u = e, e = V2 (u, e); that is, u = V2 (u, u), a second-order ˙ ˙ ¨ ˙ equation in the standard sense. This local computation also shows that the base integral curve uniquely determines an integral curve of V through a given initial condition in T Q. ........................... 15 July 1998—18h02 ...........................

197. 7.3 Euler–Lagrange Equations 183 The Euler–Lagrange Equations. From the point of view of Lagrangian vector fields, the main result concerning the Euler–Lagrange equations is the following. Theorem 7.3.3. Let Z be a Lagrangian system for L and suppose Z is a second-order equation. Then in a chart U × E, an integral curve (u(t), v(t)) ∈ U × E of Z satisfies the Euler–Lagrange equations; that is: du(t) = v(t), dt d D2 L(u(t), v(t)) · w = D1 L(u(t), v(t)) · w (7.3.7) dt for all w ∈ E. In finite dimensions, the Euler–Lagrange equations take the form dq i = qi , ˙ dt d ∂L ∂L = i, i = 1, . . . , n. (7.3.8) dt ∂ qi ˙ ∂q If L is regular, that is, ΩL is (weakly) nondegenerate, then Z is automatically second-order and if it is strongly nondegenerate, then d2 u dv = = [D2 D2 L(u, v)]−1 (D1 L(u, v) − D1 D2 L(u, v) · v), (7.3.9) dt2 dt or in finite dimensions ∂L ∂2L q j = Gij ¨ i − j i qj , ˙ i, j = 1, . . . , n, (7.3.10) ∂q ∂q ∂ q ˙ where [Gij ] is the inverse of the matrix (∂ 2 L/∂q i ∂ q j ). Thus u(t) and q i (t) ˙ are base integral curves of the Lagrangian vector field Z if and only if they satisfy the Euler–Lagrange equations. Proof. From the definition of the energy E we have the local expression DE(u, e) · (e1 , e2 ) = D1 (D2 L(u, e) · e) · e1 + D2 (D2 L(u, e) · e) · e2 − D1 L(u, e) · e1 (7.3.11) (the term D2 L(u, e) · e2 has cancelled). Locally, we may write Z(u, e) = (u, e, Y1 (u, e), Y2 (u, e)). Using formula (7.2.8) for ΩL the condition (7.3.5) on Z may be written D1 D2 L(u, e) · Y1 (u, e)) · e1 − D1 (D2 L(u, e) · e1 ) · Y1 (u, e) + D2 D2 L(u, e) · Y1 (u, e) · e2 − D2 D2 L(u, e) · e1 · Y2 (u, e) = D1 (D2 L(u, e) · e) · e1 − D1 L(u, e) · e1 + D2 D2 L(u, e) · e · e2 . (7.3.12) ........................... 15 July 1998—18h02 ...........................

198. 184 7. Lagrangian Mechanics Thus, if ΩL is a weak symplectic form, then D2 D2 L(u, e) is weakly nondegenerate, so setting e1 = 0 we get Y1 (u, e) = e; that is, Z is a second- order equation. In any case, if we assume that Z is second-order, condition (7.3.12) becomes D1 L(u, e) · e1 = D1 (D2 L(u, e) · e1 ) · e + D2 D2 L(u, e) · e1 · Y2 (u, e) (7.3.13) for all e1 ∈ E. If (u(t), v(t)) is an integral curve of Z and using dots to denote time differentiation, then u = v and u = Y2 (u, v), so (7.3.13) ˙ ¨ becomes D1 L(u, u) · e1 = D1 (D2 L(u, u) · e1 ) · u + D2 D2 L(u, u) · e1 · u ˙ ˙ ˙ ˙ ¨ d = D2 L(u, u) · e1 ˙ (7.3.14) dt by the chain rule. The last statement follows by using the chain rule on the left-hand side of Lagrange’s equation and using nondegeneracy of L to solve for v, that ˙ is, q j . ¨ Exercises 7.3-1. Give an explicit example of a degenerate Lagrangian L that has a second-order Lagrangian system Z. 7.3-2. Check directly that the validity of the expression (7.3.8) is coordinate independent. In other words, verify directly that the form of the Euler–Lagrange equations does not depend on the local coordinates chosen to describe them. 7.4 Hyperregular Lagrangians and Hamiltonians Above we said that a smooth Lagrangian L : T Q → R is hyperregular if FL : T Q → T ∗ Q is a diffeomorphism. From (7.2.8) or (7.2.11) it follows that the symmetric bilinear form D2 D2 L(u, e) is strongly nondegenerate. As before, let πQ : T ∗ Q → Q and τQ : T Q → Q denote the canonical projections. Proposition 7.4.1. Let L be a hyperregular Lagrangian on T Q and let H = E ◦ (FL)−1 ∈ F(T ∗ Q), where E is the energy of L. Then the La- grangian vector field Z on T Q and the Hamiltonian vector field XH on T ∗ Q are FL–related, that is, (FL)∗ XH = Z. ........................... 15 July 1998—18h02 ...........................

199. 7.4 Hyperregular Lagrangians and Hamiltonians 185 Furthermore, if c(t) is an integral curve of Z and d(t) an integral curve of XH with FL(c(0)) = d(0), then FL(c(t)) = d(t) and (τQ ◦ c)(t) = (πQ ◦ d)(t). The curve (τQ ◦c)(t) is called the base integral curve of c(t) and similarly (πQ ◦ d)(t) is the base integral curve of d(t). Proof. For v ∈ T Q and w ∈ Tv (T Q), we have Ω(FL(v))(Tv FL(Z(v)), Tv FL(w)) = ((FL)∗ Ω)(v)(Z(v), w) = ΩL (v)(Z(v), w) = dE(v) · w = d(H ◦ FL)(v) · w = dH(FL(v)) · Tv FL(w) = Ω(FL(v))(XH (FL(v)), Tv FL(w)), so that by weak nondegeneracy of Ω and the fact that Tv FL is an isomorphism, it follows that Tv FL(Z(v)) = XH (FL(v)). Thus T FL ◦ Z = XH ◦ FL, that is, Z = (FL)∗ XH . If ϕt denotes the flow of Z and ψt the flow of XH , the relation Z = (FL)∗ XH is equivalent to FL ◦ ϕt = ψt ◦ FL. Thus, if c(t) = ϕt (v), then FL(c(t)) = ψt (FL(v)) is an integral curve of XH which at t = 0 passes through FL(v) = FL(c(0)), whence ψt (FL(v)) = d(t) by uniqueness of integral curves of smooth vector fields. Finally, since τQ = πQ ◦ FL, we get (τQ ◦ c)(t) = (πQ ◦ FL ◦ c)(t) = (πQ ◦ d)(t). The Action. We claim that the action A of L is related to the Lagrangian vector field Z of L by A(v) = ΘL (v), Z(v) , v ∈ T Q. (7.4.1) We prove this formula under the assumption that Z is a second-order equation, even if L is not regular. In fact, ΘL (v), Z(v) = ((FL)∗ Θ)(v), Z(v) = Θ(FL(v)), Tv FL(Z(v)) = FL(v), T πQ · Tv FL(Z(v)) = FL(v), Tv (πQ ◦ FL)(Z(v)) = FL(v), Tv τQ (Z(v)) = FL(v), v = A(v), ........................... 15 July 1998—18h02 ...........................

200. 186 7. Lagrangian Mechanics by definition of a second-order equation and the definition of the action. If L is hyperregular and H = E ◦ (FL)−1 , then A ◦ (FL)−1 = Θ, XH . (7.4.2) Indeed, by (7.4.1), the properties of push-forward, and the previous proposition, we have A ◦ (FL)−1 = (FL)∗ A = (FL)∗ ( ΘL , Z ) = (FL)∗ ΘL , (FL)∗ Z = Θ, XH . If H : T ∗ Q → R is a smooth Hamiltonian, the function G : T ∗ Q → R given by G = Θ, XH is called the action of H. Thus, (7.4.2) says that the push-forward of the action A of L equals the action G of H = E ◦ (FL)−1 . Hyperregular Hamiltonians. A Hamiltonian H is called hyperregular if FH : T ∗ Q → T Q, defined by d FH(α) · β = H(α + sβ), (7.4.3) ds s=0 ∗ where α, β ∈ Tq Q, is a diffeomorphism; here we must assume that either ∗∗ the model space E of Q is reflexive so that Tq Q = Tq Q for all q ∈ Q or, ∗∗ what is more reasonable, that FH(α) lies in Tq Q ⊂ Tq Q. As in the case of Lagrangians, hyperregularity of H implies the strong nondegeneracy of D2 D2 H(u, α) and the curve s → α+sβ appearing in (7.4.3) can be replaced ∗ by an arbitrary smooth curve α(s) in Tq Q such that α(0) = α and α (0) = β. Proposition 7.4.2. (i) Let H ∈ F(T ∗ Q) be a hyperregular Hamiltonian and define E = H ◦ (FH)−1 , A = G ◦ (FH)−1 , and L = A − E ∈ F(T Q). Then L is a hyperregular Larangian and FL = FH −1 . Furthermore, A is the action of L and E the energy of L. (ii) Let L ∈ F(T Q) be a hyperregular Lagrangian and define H = E ◦ (FL)−1 . Then H is a hyperregular Hamiltonian and FH = (FL)−1 . Proof. (i) Locally G(u, α) = α, D2 H(u, α) , so that A(u, D2 H(u, α)) = (A ◦ FH)(u, α) = G(u, α) = α, D2 H(u, α) , whence (L ◦ FH)(u, α) = L(u, D2 H(u, α)) = α, D2 H(u, α) − H(u, α). ........................... 15 July 1998—18h02 ...........................

201. 7.4 Hyperregular Lagrangians and Hamiltonians 187 Let e = D2 (D2 H(u, α)) · β and let e(s) = D2 H(u, α + sβ) be a curve which at s = 0 passes through e(0) = D2 H(u, α) and whose derivative at s = 0 equals e (0) = D2 (D2 H(u, α)) · β = e. Therefore, (FL ◦ FH)(u, α), e = FL(u, D2 H(u, α)), e d d = L(u, e(s)) = L(u, D2 H(u, α + sβ)) dt s=0 dt s=0 d = [ α + sβ, D2 H(u, α + sβ) − H(u, α + sβ)] dt s=0 = α, D2 (D2 H(u, α)) · β = α, e . Since D2 D2 H(u, α) is strongly nondegenerate this implies that e ∈ E is arbitrary and hence FL ◦ FH = identity. Since FH is a diffeomorphism, this says that FL = (FH)−1 and hence that L is hyperregular. To see that A is the action of L note that since FH −1 = FL we have by definition of G A = G ◦ (FH)−1 = Θ, XH ◦ FL, which by (7.4.2) implies that A is the action of L. Therefore, E = A − L is the energy of L. (ii) Locally, since we define H = E ◦ (FL)−1 , we have (H ◦ FL)(u, e) = H(u, D2 L(u, e)) = A(u, e) − L(u, e) = D2 L(u, e) · e − L(u, e) and proceed as before. Let α = D2 (D2 L(u, e)) · f, where f ∈ E, and α(s) = D2 L(u, e + sf ); then α(0) = D2 L(u, e), and α (0) = α, so that α, (FH ◦ FL)(u, e) = α, FH(u, D2 L(u, e)) d = H(u, α(s)) ds s=0 d = H(u, D2 L(u, e + sf )) ds s=0 d = [ D2 L(u, e + sf ), e + sf − L(u, e + sf )] ds s=0 = D2 (D2 L(u, e)) · f, e = α, e , ........................... 15 July 1998—18h02 ...........................

202. 188 7. Lagrangian Mechanics which shows, by strong nondegeneracy of D2 D2 L, that FH ◦ FL = identity. Since FL is a diﬀeomorphism it follows that FH = (FL)−1 and H is hyperregular. The main result is summarized in the following. Theorem 7.4.3. Hyperregular Lagrangians L ∈ F(T Q) and hyperregular Hamiltonians H ∈ F(T ∗ Q) correspond in a bijective manner by the preceding constructions. The following diagram commutes: T FH E T (T Q) T (T ∗ Q) ' T T FL T R XH s d XE G d A d FL d ∗ ' L E TQ E R T Q FH d d H d E ‚ d © R Proof. Let L be a hyperregular Lagrangian and let H be the associated Hamiltonian which is hyperregular, that is, H = E ◦ (FL)−1 = (A − L) ◦ (FL)−1 = G − L ◦ FH by Propositions 7.4.1 and 7.4.2. From H we construct a Lagrangian L by L = G ◦ (FH)−1 − H ◦ (FH)−1 = G ◦ (FH)−1 − (G − L ◦ FH) ◦ (FH)−1 = L. Conversely, if H is a given hyperregular Hamiltonian, then the associated Lagrangian L is hyperregular and is given by L = G ◦ (FH)−1 − H ◦ (FH)−1 = A − H ◦ FL. Thus, the corresponding hyperregular Hamiltonian induced by L is H = E ◦ (FL)−1 = (A − L) ◦ (FL)−1 = A ◦ (FL)−1 − (A − H ◦ FL) ◦ (FL)−1 = H. The commutativity of the two diagrams is now a direct consequence of the above and Propositions 7.4.1 and 7.4.2. ........................... 15 July 1998—18h02 ...........................

203. , 7.4 Hyperregular Lagrangians and Hamiltonians 189 Neighborhood Theorem for Regular Lagrangians. We now prove an important theorem for regular Lagrangians that concerns the structure of solutions near a given one. Definition 7.4.4. Let q(t) be a given solution of the Euler–Lagrange , equations, t1 ≤ t ≤ t2 . Let q 1 = q t1 and q 2 = q t2 . We say that q(t) is a nonconjugate solution if there is a neighborhood U of the curve q(t) and neighborhoods U1 ⊂ U of q 1 and U2 ⊂ U of q 2 such that for all q1 ∈ U1 and q2 ∈ U2 and t1 close to t1 , t2 close to t2 , there exists a unique solution q(t), t1 ≤ t ≤ t2 of the Euler–Lagrange equations satisfying the following conditions: q (t1 ) = q1 , q (t2 ) = q2 and q(t) ∈ U. See Figure 7.4.1. U2 U q 2 _ q 2 _ q (t) q (t) q 1 U1 _ q 1 Figure 7.4.1. Neighborhood Theorem To determine conditions guaranteeing that a solution is nonconjugate, ˙ ˙ we shall use the following observation. Let v 1 = q (t1 ) and v 2 = q (t2 ). Let Ft be the flow of the Euler–Lagrange equations on T Q. By construction of Ft (q, v), we have Ft2 (q 1 , v 1 ) = (q 2 , v 2 ) . Next, we attempt to apply the implicit function theorem to the flow map. We want to solve (πQ ◦ Ft2 ) (q1 , v1 ) = q2 for v1 , where we regard q1 , t1 , t2 as parameters. To do this, we form the linearization w2 := Tv1 (πQ ◦ Ft2 ) (q 1 , v 1 ) · w1 . We require that w1 → w2 is invertible. The right-hand side of this equation suggests forming the curve w(t) := Tv1 πQ Ft (q 1 , v 1 ) · w1 . (7.4.4) ........................... 15 July 1998—18h02 ...........................

204. 190 7. Lagrangian Mechanics which is the solution of the linearized, or first variation, equation of the Euler–Lagrange equations satisfied by Ft (q 1 , v 1 ). Let us work out the equation satisfied by w(t) := Tv1 πQ Ft (q 1 , v 1 ) · w1 in coordinates. Start with a solution q(t) of the Euler–Lagrange equations d ∂L ∂L − i = 0. dt ∂ q i ˙ ∂q Given the curve of initial conditions ε → (q1 , v1 + εw1 ), we get corresponding solutions (qε (t), qε (t)), whose derivative with respect to ε we denote ˙ (u(t), u(t)). Differentiation of the Euler–Lagrange equations with respect ˙ to ε gives d ∂2L ∂2L ∂2L ∂2L i ∂ qj · uj + i j · uj ˙ − i ∂q j · uj − i j · uj = 0 (7.4.5) dt ∂q ˙ ˙ ∂ q ∂q ˙ ∂q ∂q ∂ q ˙ which is a second-order equation for uj . This equation evaluated along q(t) is called the Jacobi equation along q(t). This equation, taken from q(t1 ) to q(t2 ) with initial conditions u(t1 ) = 0 and u(t1 ) = w1 , ˙ defines the desired linear map w1 → w2 : that is, w2 = u(t2 ). ˙ Theorem 7.4.5. Assume L is a regular Lagrangian. If the linear map w1 → w2 is an isomorphism, then q(t) is nonconjugate. Proof. This follows directly from the implicit function theorem. Under the hypothesis that w1 → w2 is invertible, there are neighborhoods U1 of q 1 , U2 of q 2 and neighborhoods of t1 and t2 as well as a smooth function v1 = v1 (t1 , t2 , q1 , q2 ) defined on the product of these four neighborhoods such that (πQ ◦ Ft2 ) (q1 , v1 (t1 , t2 , q1 , q2 )) = q2 (7.4.6) is an identity. Then q(t) := (πQ ◦ Ft )(q1 , v1 (t1 , t2 , q1 , q2 )) is a solution of the Euler–Lagrange equations with initial conditions (q1 , v1 (t1 , t2 , q1 , q2 )) at t = t1 . Moreover, q(t2 ) = q2 by (7.4.6). ........................... 15 July 1998—18h02 ...........................

205. 7.5 Geodesics 191 If q1 and q2 are close and if t2 is not much different from t1 , then by continuity, u(t) is approximately constant over [t1 , t2 ], so that ˙ w2 = u(t2 ) = (t2 − t1 )u(t1 ) + O(t2 − t1 )2 = (t2 − t1 )w1 + O(t2 − t1 )2 . ˙ ˙ Thus, in these circumstances, the map w1 → w2 is invertible. Therefore, we get Corollary 7.4.6. Let L : T Q × R → R be a given C 2 regular Lagrangian and let vq ∈ T Q and t1 ∈ R. Then the solution of the Euler–Lagrange equations with initial condition vq at t = t1 is nonconjugate for a sufficiently small time interval [t1 , t2 ]. The term “nonconjugate” comes from the study of geodesics, which are considered in the next section. Exercises 7.4-1. Write down the Lagrangian and the equations of motion for a spherical pendulum with S 2 as configuration space. Convert the equations to Hamiltonian form using the Legendre transformation. Find the conservation law corresponding to angular momentum about the axis of gravity by “bare hands” methods. 7.4-2. Let L(q, q) = 1 m(q)q 2 − V (q) on T R, where m(q) 0 and V (q) ˙ 2 ˙ are smooth. Show that any two points q1 , q2 ∈ R can be joined by a solution of the Euler–Lagrange equations. (Hint: Consider the energy equation.) 7.5 Geodesics Let Q be a weak pseudo-Riemannian manifold whose metric evaluated at q ∈ Q is denoted interchangeably by · , · or g(q) or gq . Consider on T Q the Lagrangian given by the kinetic energy of the metric, that is, 1 L(v) = 2 v, v q , (7.5.1) or in finite dimensions L(v) = 1 gij v i v j . 2 (7.5.2) The fiber derivative of L is given for v, w ∈ Tq Q by FL(v) · w = v, w (7.5.3) or in finite dimensions by FL(v) · w = gij v i wj i.e., pi = gij q j . ˙ (7.5.4) ........................... 15 July 1998—18h02 ...........................

206. 192 7. Lagrangian Mechanics From this equation we see that in any chart U for Q, D2 D2 L(q, v) · (e1 , e2 ) = e1 , e2 q , where , q denotes the inner product on E induced by the chart. Thus, L is automatically weakly nondegenerate. Note that the action is given by A = 2L, so E = L. The Lagrangian vector field Z in this case is denoted by S : T Q → T 2 Q and is called the Christoffel map or geodesic spray of the metric , q . Thus, S is a second-order equation and hence has a local expression of the form S(q, v) = ((q, v), (v, γ(q, v))) (7.5.5) in a chart on Q. To determine the map γ : U × E → E from Lagrange’s equations, note that D1 L(q, v) · w = 1 Dq v, v 2 q ·w and D2 L(q, v) · w = v, w q (7.5.6) so that the Euler–Lagrange equations (7.3.7) are q = v, ˙ (7.5.7) d ( v, w q ) = 1 Dq v, v 2 q · w. (7.5.8) dt Keeping w fixed and expanding the left-hand side of (7.5.8) yields Dq v, w q · q + v, w ˙ ˙ q . (7.5.9) Taking into account q = v, we get ˙ q, w ¨ q = 1 Dq v, v 2 q · w − Dq v, w q · v. (7.5.10) Hence γ : U × E → E is defined by the equality γ(q, v), w q = 1 Dq v, v 2 q · w − Dq v, w q · v; (7.5.11) note that γ(q, v) is a quadratic form in v. If Q is finite dimensional, we define the Christoffel symbols Γi by putting jk γ i (q, v) = −Γi (q)v j v k jk (7.5.12) and demanding Γi = Γi . With this notation, the relation (7.5.11) is jk kj equivalent to 1 ∂gjk j k l ∂gjl j l k −gil Γi v j v k wl = jk v v w − v wv . (7.5.13) 2 ∂q l ∂q k ........................... 15 July 1998—18h02 ...........................

207. 7.5 Geodesics 193 Taking into account the symmetry of Γi , this gives jk 1 hl ∂gjl ∂gkl ∂gjk Γh = jk g k + j − . (7.5.14) 2 ∂q ∂q ∂q l In infinite dimensions, since the metric , is only weakly nondegenerate (7.5.11) guarantees the uniqueness of γ but not its existence. It exists whenever the Lagrangian vector field S exists. The integral curves of S projected to Q are called geodesics of the metric g. By (7.5.5), their basic governing equation has the local expression q = γ(q, q), ¨ ˙ (7.5.15) which, in finite dimensions, reads q i + Γi q j q k = 0, ¨ jk ˙ ˙ (7.5.16) where i, j, k = 1, . . . , n and, as usual, there is a sum on j and k. Note that the definition of γ makes sense both in the finite- and infinite-dimensional case, where as the Christoffel symbols Γi are defined only for finite- jk dimensional manifolds. Working intrinsically with g provides a way to deal with geodesics of weak Riemannian (and pseudo-Riemannian) metrics on infinite-dimensional manifolds. Taking the Lagrangian approach as basic, we see where the Γi live jk as geometric objects: in T (T Q) since they encode the principal part of the Lagrangian vector field Z. If one writes down the transformation properties of Z on T (T Q) in natural charts, the classical transformation rule for the Γi results: jk k ∂q p ∂q m r ∂q k ∂q k ∂ 2 q l Γij = Γpm r + , (7.5.17) ∂q i ∂q j ∂q ∂q l ∂q i ∂q j where (q 1 , . . . , q n ), (q 1 , . . . , q n ) are two different coordinate systems on an open set of Q. We leave this calculation to the reader. The Lagrangian approach leads naturally to invariant manifolds for the geodesic flow. For example, for each real e 0, let Σe = {v ∈ T Q | v = e} √ be the pseudo-sphere bundle of radius e in T Q. Then Σe is a smooth submanifold of T Q invariant under the geodesic flow. Indeed, if we show that Σe is a smooth submanifold, its invariance under the geodesic flow, that is, under the flow of Z, follows by conservation of energy. To show that Σe is a smooth submanifold we prove that e is a regular value of L for e 0. This is done locally by (7.5.6) DL(u, v) · (w1 , w2 ) = D1 L(u, v) · w1 + D2 L(u, v) · w2 = 1 Du v, v 2 u · w1 + v, w2 u = v, w2 u , (7.5.18) ........................... 15 July 1998—18h02 ...........................

208. 194 7. Lagrangian Mechanics since v, v = 2e = constant. By weak nondegeneracy of the pseudo-metric , , this shows that DL(u, v) : E × E → R is a surjective linear map, that is, e is a regular value of L. Convex Neighborhoods and Conjugate Points. We proved in the last section that short arcs of solutions of the Euler–Lagrange equations are nonconjugate. In the special case of geodesics one can do somewht better by exploiting the fact, evident from the quadratic nature of (7.5.16), that if q(t) is a solution and α 0, then so is q(αt), so one can “rescale” solutions simply by changing the size of the initial velocity. One finds that locally there are convex neighborhoods; that is, neighborhoods U such that for any q1 , q2 ∈ U , there is a unique geodesic (up to a scaling) joining q1 , q2 and lying ind U . In Riemannian geometry there is another important result, the Hopf–Rinow Theorem stating that any two points (in the same connected component) can be joined by some geodesic. As one follows a geodesic from a given point, there is a first point after which nearby geodesics fail to be unique. These are conjugate points. They are the zeros of the Jacobi equation discussed earlier. For example, on a great circle on a sphere, pairs of antipodal points are conjugate. In certain circumstances one can “reduce” the Euler–Lagrange problem to one of geodescis: see the discussion fo the Jacobi metric in §7.7. Covariant derivatives. We now reconcile the above approach to geodesics via Lagrangian systems to a common approach in differential geometry. Define the covariant derivative : X(Q) × X(Q) → X(Q); (X, Y ) → XY locally by ( XY )(u) = −γ(u)(X(u), Y (u)) + DY (u) · X(u), (7.5.19) where X, Y are the local representatives of X and Y and γ(u) : E × E → E denotes the symmetric bilinear form defined by the polarization of γ(u, v), which is a quadratic form in v. In local coordinates, the preceding equation becomes ∂ ∂Y k ∂ XY = X j Y k Γi jk i + Xj j . (7.5.20) ∂q ∂q ∂q k It is straightforward to check that this definition is chart independent and that satisfies the following conditions: (i) is R–bilinear; (ii) for f : Q → R, fX Y =f XY and XfY =f XY + X[f ]Y ; and ........................... 15 July 1998—18h02 ...........................

209. 7.5 Geodesics 195 (iii) for vector fields X and Y , ( XY − Y X)(u) = DY (u) · X(u) − DX(u) · Y (u) = [X, Y ](u). (7.5.21) In fact, these three properties characterize covariant derivative operators. The particular covariant derivative determined by (7.5.14) is called the Levi–Civita covariant derivative. If c(t) is a curve in Q and X ∈ X(Q), the covariant derivative of X along c is defined by DX = u X, (7.5.22) Dt where u is a vector field coinciding with c(t) at c(t). This is possible since, by ˙ (7.5.19) or (7.5.20), X Y depends only on the point values of X. Explicitly, in a local chart, we have DX d (c(t)) = −γc(t) (u(c(t)), X(c(t))) + X(c(t)), (7.5.23) Dt dt which shows that DX/Dt depends only on c(t) and not on how c(t) is ˙ ˙ extended to a vector field. In finite dimensions, i DX d i = Γi (c(t))cj (t)X k (c(t)) + jk ˙ X (c(t)). (7.5.24) Dt dt The vector field X is called autoparallel or parallel transported along c if DX/Dt = 0. Thus c is autoparallel along c if and only if ˙ c(t) − γ(t)(c(t), c(t)) = 0, ¨ ˙ ˙ that is, c(t) is a geodesic. In finite dimensions, this reads ci + Γi cj ck = 0. ¨ jk ˙ ˙ Exercises 7.5-1. Consider the Lagrangian 1 2 L (x, y, z, x, y, z) = ˙ ˙ ˙ 1 2 x2 + y 2 + z 2 − ˙ ˙ ˙ 1 − x2 + y 2 + z 2 2 for a particle in R3 . Let γ (t) be the curve in R3 obtained by solving the Euler–Lagrange equations for L with the initial conditions x0 , v0 = γ (0). ˙ Show that lim γ (t) →0 is a great circle on the two-sphere S 2 , provided that x0 has length one and that x0 · v0 = 0. 7.5-2. Write out the geodesic equations in terms of q i and pi and check directly that Hamilton’s equations are satisfied. ........................... 15 July 1998—18h02 ...........................

210. 196 7. Lagrangian Mechanics 7.6 The Kaluza–Klein Approach to Charged Particles In §6.7 we studied the motion of a charged particle in a magnetic field as a Hamiltonian system. Here we show that this description is the reduction of a larger and, in some sense, simpler system called the Kaluza–Klein system.1 Physically, we are motivated as follows: since charge is a basic conserved quantity, we would like to introduce a new cyclic variable whose conjugate momentum is the charge.2 For a charged particle, the resultant system is in fact geodesic motion! Recall from §6.7 that if B = × A is a given magnetic field on R3 , then with respect to canonical variables (q, p), the Hamiltonian is 1 e 2 H(q, p) = p− A . (7.6.1) 2m c First we claim that we can obtain (7.6.1) via the Legendre transform if we choose e 2 ˙ 2 L(q, q) = 1 m q + A · q. ˙ ˙ (7.6.2) c Indeed, in this case, ∂L e p= ˙ = mq + A (7.6.3) ˙ ∂q c and H(q, p) = p · q − L(q, q) ˙ ˙ e 2 e = mq + A · q − 1 m q ˙ ˙ 2 ˙ − A·q ˙ c c 1 e 2 = 1m q = 2 ˙ 2 p− A . (7.6.4) 2m c Thus, the Euler–Lagrange equations for (7.6.2) reproduce the equations for a particle in a magnetic field.3 Let the configuration space be QK = R3 × S 1 (7.6.5) 1 After learning reduction theory (see Abraham and Marsden [1978] or Marsden [1992]), the reader can revisit this construction, but here all the constructions are done directly. 2 This process is applicable to other situations as well; for example, in fluid dynamics one can profitably introduce a variable conjugate to the conserved mass density or entropy; see Marsden, Ratiu, and Weinstein [1984a,b]. 3 If an electric field E = − ϕ is also present as well, one simply subtracts eϕ from L, treating eϕ as a potential energy, as in the next section. ........................... 15 July 1998—18h02 ...........................

211. 7.6 The Kaluza–Klein Approach to Charged Particles 197 with variables (q, θ), define A = A , a one-form on R3 , and consider the one-form ω = A + dθ (7.6.6) on QK called the connection one-form. Let the Kaluza–Klein La- grangian be defined by 2 ˙ ˙ LK (q, q, θ, θ) = 1 m q ˙ 2 + 1 ˙ ˙ ω, (q, q, θ, θ) 2 2 = 1m q ˙ 2 ˙ ˙ + 1 (A · q + θ)2 . (7.6.7) 2 2 The corresponding momenta are ˙ ˙ ˙ p = mq + (A · q + θ)A (7.6.8) and ˙ ˙ p = A · q + θ. (7.6.9) ˙ ˙ Since LK is quadratic and positive-definite in q and θ, the Euler–Lagrange equations are the geodesic equations on R × S for the metric for which 3 1 LK is the kinetic energy. Since p is constant in time, as can be seen from the ˙ Euler–Lagrange equation for (θ, θ), we can define the charge e by setting p = e/c; (7.6.10) then (7.6.8) coincides with (7.6.3). The corresponding Hamiltonian on T ∗ QK endowed with the canonical symplectic form is 1 HK (q, p, θ, p) = p − pA 2 + 1 p2 . 2 (7.6.11) 2m With (7.6.10), (7.6.11) differs from (7.6.1) by the constant p2 /2. These constructions generalize to the case of a particle in a Yang–Mills field where ω becomes the connection of a Yang–Mills field and its curvature measures the field strength which, for an electromagnetic field, reproduces the relation B = × A. Also, the possibility of putting the interaction in the Hamiltonian, or via a momentum shift, into the symplectic structure, also generalizes. We refer to Wong [1970], Sternberg [1977], Weinstein [1978], and Montgomery [1984] for details and further references. Finally, we remark that the relativistic context is the most natural to introduce the full electromagnetic field. In that setting the construction we have given for the magnetic field will include both electric and magnetic effects. Consult Misner, Thorne, and Wheeler [1973] for additional information. ........................... 15 July 1998—18h02 ...........................

212. 198 7. Lagrangian Mechanics Exercises 7.6-1. The bob on a spherical pendulum has a charge e, m, and moves under the influence of a constant gravitational field with acceleration g, and a magnetic field B. Write down the Lagrangian, the Euler–Lagrange equations, and the variational principle for this system. Transform the system to Hamiltonian form. Find a conserved quantity if the field B is symmetric about the axis of gravity. 7.7 Motion in a Potential Field We now generalize geodesic motion to include potentials V : Q → R. Recall that the gradient of V is the vector field grad V = V defined by the equality grad V (q), v q = dV (q) · v, (7.7.1) for all v ∈ Tq Q. In finite dimensions, this definition becomes ∂V (grad V )i = g ij . (7.7.2) ∂q j Define the (weakly nondegenerate) Lagrangian L(v) = 1 v, v q − V (q). 2 A computation similar to the one in §7.5 shows that the Euler–Lagrange equations are q = γ(q, q) − grad V (q), ¨ ˙ (7.7.3) or in finite dimensions ∂V q i + Γi q j q k + g il ¨ jk ˙ ˙ = 0. (7.7.4) ∂q l The action of L is given by A(v) = v, v q , (7.7.5) so that the energy is E(v) = A(v) − L(v) = 1 2 v, v q + V (q). (7.7.6) The equations (7.7.3) written as q = v, v = γ(q, v) − grad V (q) ˙ ˙ (7.7.7) are thus Hamiltonian with Hamiltonian function E with respect to the symplectic form ΩL . ........................... 15 July 1998—18h02 ...........................

213. 7.7 Motion in a Potential Field 199 Invariant Form. There are several ways to write equations (7.7.7) in invariant form. Perhaps the simplest is to use the language of covariant derivatives from the last section and to write Dc ˙ =− V (7.7.8) Dt or, what is perhaps better, Dc ˙ g = −dV (7.7.9) Dt where g : T Q → T ∗ Q is the map associated to the Riemmanian metric. This last equation is the geometric way of writing ma = F. Another method uses the following terminology: Definition 7.7.1. Let v, w ∈ Tq Q. The vertical lift of w with respect to v is defined by d ver(w, v) = (v + tw) ∈ Tv (T Q). dt t=0 The horizontal part of a vector U ∈ Tv (T Q) is Tv τQ (U ) ∈ Tq Q. A vector field is called vertical if its horizontal part is zero. In charts, if v = (u, e), w = (u, f ), and U = ((u, e), (e1 , e2 )), the definition says that ver(w, v) = ((u, e), (0, f )) and Tv τQ (U ) = (u, e1 ). Thus, U is vertical iff e1 = 0. Thus, any vertical vector U ∈ Tv (T Q) is the vertical lift of some vector w (which in a natural local chart is (u, e2 )) with respect to v. If S denotes the geodesic spray of the metric , on T Q, equations (7.7.7) say that the Lagrangian vector field Z defined by L(v) = 1 v, v q − V (q), 2 where v ∈ Tq Q, is given by Z = S − ver( V ), (7.7.10) that is, Z(v) = S(v) − ver(( V )(q), v). (7.7.11) Remarks. In general, there is no canonical way to take the vertical part of a vector U ∈ Tv (T Q) without extra structure. Having such a structure is what one means by a connection. In case Q is pseudo-Riemannian, such a projection can be constructed in the following manner. Suppose, in natural charts, that U = ((u, e), (e1 , e2 )). Define Uver = ((u, e), (0, γ(u)(e1 , e2 ) + e2 )) ........................... 15 July 1998—18h02 ...........................

214. 200 7. Lagrangian Mechanics where γ(u) is the bilinear symmetric form associated to the quadratic form γ(u, e) in e. We conclude with some miscellaneous remarks connecting motion in a potential field with geodesic motion. We confine ourselves to the finite– dimensional case for simplicity. Definition 7.7.2. Let g = , be a pseudo-Riemannian metric on Q and let V : Q → R be bounded above. If e V (q) for all q ∈ Q define the Jacobi metric ge by ge = (e − V )g, that is, ge (v, w) = (e − V (q)) v, w for all v, w ∈ Tq Q. Theorem 7.7.3. Let Q be finite dimensional. The base integral curves of the Lagrangian L(v) = 1 v, v − V (q) with energy e are the same as 2 geodesics of the Jacobi metric with energy 1, up to a reparametrization. The proof is based on the following of separate interest. Proposition 7.7.4. Let (P, Ω) be a (finite–dimensional) symplectic manifold, H, K ∈ F(P ), and assume that Σ = H −1 (h) = K −1 (k) for h, k ∈ R regular values of H and K, respectively. Then the integral curves of XH and XK on the invariant submanifold Σ of both XH and XK coincide up to a reparametrization. Proof. From Ω(XH (z), v) = dH(z) · v, we see that XH (z) ∈ (ker dH(z))Ω = (Tz Σ)Ω , the symplectic orthogonal complement of Tz Σ. Since dim P = dim Tz Σ + dim(Tz Σ)Ω (see §2.3) and since Tz Σ has codimension one, (Tz Σ)Ω has dimension one. Thus, the nonzero vectors XH (z) and XK (z) are multiples of each other at every point z ∈ Σ, that is, there is a smooth nowhere vanishing function λ : Σ → R such that XH (z) = λ(z)XK (z) for all z ∈ Σ. Let c(t) be the integral curve of XK with initial condition c(0) = z0 ∈ Σ. The function ϕ ϕ → 0 dt/(λ ◦ c)(t) is a smooth monotone function and therefore has an inverse t → ϕ(t) . If d(t) = (c ◦ ϕ)(t), then d(0) = z0 and 1 d (t) = ϕ (t)c (ϕ(t)) = XK (c(ϕ(t))) = (λ ◦ c)(ϕ)XK (d(t)) t (ϕ) = λ(d(t))XK (d(t)) = XH (d(t)) that is, the integral curve of XH through z0 is obtained by reparametrizing the integral curve of XK through z0 . ........................... 15 July 1998—18h02 ...........................

215. 7.8 The Lagrange–d’Alembert Principle 201 Proof of Theorem 7.7.3. Let H be the Hamiltonian for L, namely 1 2 H(q, p) = 2 p + V (q) and He be that for the Jacobi metric: He (q, p) = 1 (e − V (q))−1 p 2 . 2 The factor (e − V (q))−1 occurs because the inverse metric is used for the momenta. Clearly, H = e defines the same set as He = 1, so the result follows from Proposition 7.7.4 if we show that e is a regular value of H and 1 is a regular value of He . Note that if (q, p) ∈ H −1 (e), then p = 0 since e V (q) for all q ∈ Q. Therefore, FH(q, p) = 0 for any (q, p) ∈ H −1 (e) and hence dH(q, p) = 0, that is, e is a regular value of H. Since 1 FHe (q, p) = ˙ (e − V (q))−1 FH(q, p), 2 this also shows that FHe (q, p) = 0 for all (q, p) ∈ H −1 (e) = He (1) −1 and thus 1 is a regular value of He . 7.8 The Lagrange–d’Alembert Principle In this section we study a generalization of Lagrange’s equations for mechanical systems with exterior forces. A special class of such forces is dissipative forces, which will be studied at the end of this section. Force Fields. Let L : T Q → R be a Lagrangian function, let Z be the Lagrangian vector field associated to L, assumed to be a second-order equation, and denote by τQ : T Q → Q the canonical projection. Recall that a vector field Y on T Q is called vertical if T τQ ◦ Y = 0. Such a vector field Y defines a one-form ∆Y on T Q by contraction with ΩL : ∆Y = −iY ΩL = Y ΩL . Proposition 7.8.1. If Y is vertical, then ∆Y is a horizontal one- form, that is, ∆Y (U ) = 0 for any vertical vector field U on T Q. Con- versely, given a horizontal one-form ∆ on T Q, and assuming that L is regular, the vector field Y on T Q, defined by ∆ = −iY ΩL , is vertical. Proof. This follows from a straightforward calculation in local coordinates. We use the fact that a vector field Y (u, e) = (Y1 (u, e), Y2 (u, e)) is ........................... 15 July 1998—18h02 ...........................

216. 202 7. Lagrangian Mechanics vertical if and only if the first component Y1 is zero and the local formula for ΩL derived earlier: ΩL (u, e)(Y1 , Y2 ), (U1 , U2 )) = D1 (D2 L(u, e) · Y1 ) · U1 − D1 (D2 L(u, e) · U1 ) · Y1 + D2 D2 L(u, e) · Y1 · U2 − D2 D2 L(u, e) · U1 · Y2 . (7.8.1) This shows that (iY ΩL )(U ) = 0 for all vertical U is equivalent to D2 D2 L(u, e)(U2 , Y1 ) = 0. If Y is vertical, this is clearly true. Conversely if L is regular, and the last displayed equation is true, then Y1 = 0, so Y is vertical. Proposition 7.8.2. Any fiber-preserving map F : T Q → T ∗ Q over the ˜ identity induces a horizontal one-form F on T Q by ˜ F (v) · Vv = F (v), Tv τQ (Vv ) , (7.8.2) where v ∈ T Q and Vv ∈ Tv (T Q). Conversely, formula (7.8.2) defines, for ˜ any horizontal one-form F , a fiber-preserving map F over the identity. Any such F is called a force field and thus, in the regular case, any vertical vector field Y is induced by a force field. Proof. Given F , formula (7.8.2) clearly defines a smooth one-form F on ˜ T Q. If Vv is vertical, then the right-hand side of formula (7.8.2) vanishes, ˜ and so F is a horizontal one-form. Conversely, given a horizontal one-form F˜ on T Q, and given v, w ∈ Tq Q, let Vv ∈ Tv (T Q) be such that Tv τ (Vv ) = w. ˜ Then define F by formula (7.8.2); that is, F (v), w = F (v) · Vv . Since F is ˜ horizontal, we see that F is well defined, and its expression in charts shows that it is smooth. Treating ∆Y as the exterior force one-form acting on a mechanical system with a Lagrangian L, we now will write the governing equations of motion. The Lagrange–d’Alembert Principle. First, we recall the definition from Vershik and Faddeev [1981] and Wang and Krishnaprasad [1992]. Definition 7.8.3. The Lagrangian force associated with a Lagrangian L and a given second-order vector field (the ultimate equations of motion) X is the horizontal one-form on T Q defined by ΦL (X) = iX ΩL − dE. (7.8.3) Given a horizontal one-form ω (referred to as the exterior force one- form), the local Lagrange d’Alembert principle associated with the second-order vector field X on T Q states that ΦL (X) + ω = 0. (7.8.4) ........................... 15 July 1998—18h02 ...........................

217. 7.8 The Lagrange–d’Alembert Principle 203 It is easy to check that ΦL (X) is indeed horizontal if X is second-order. Conversely, if L is regular and if ΦL (X) is horizontal, then X is second- order. One can also formulate an equivalent principle in terms of variational principles. Definition 7.8.4. Given a Lagrangian L and a force field F , as defined in Proposition 7.8.2, the integral Lagrange–d’Alembert principle for a curve q(t) in Q is b b δ L(q(t), q(t)) dt + ˙ F (q(t), q(t)) · δq dt = 0, ˙ (7.8.5) a a where the variation is given by the usual expression b b ∂L i ∂L d i δ L(q(t), q(t)) dt = ˙ δq + i δq dt. a a ∂q i ∂ q dt ˙ b ∂L d ∂L = i − δq i dt. (7.8.6) a ∂q dt ∂ q i ˙ for a given variation δq (vanishing at the endpoints). The two forms of the Lagrange–d’Alembert principle are in fact equivalent. This will follow from the fact that both give the Euler–Lagrange equations with forcing in local coordinates (provided that Z is second-order). We shall see this in the following development. Proposition 7.8.5. Let the exterior force one-form ω be associated to a vertical vector field Y , that is, let ω = ∆Y = −iY ΩL . Then X = Z + Y satisfies the local Lagrange–d’Alembert principle. Conversely, if, in addition, L is regular, the only second-order vector field X satisfying the local Lagrange–d’Alembert principle is X = Z + Y . Proof. For the first part, the equality ΦL (X) + ω = 0 is a simple verification. For the converse, we already know that X is a solution, and uniqueness is guaranteed by regularity. To develop the differential equations associated to X = Z + Y , we take ω = −iY ΩL and note that, in a coordinate chart, Y (q, v) = (0, Y2 (q, v)) since Y is vertical, that is, Y1 = 0. From the local formula for ΩL , we get ω(q, v) · (u, w) = D2 D2 L(q, v) · Y2 (q, v) · u. (7.8.7) Letting X(q, v) = (v, X2 (q, v)), one finds that ΦL (X)(q, v) · (u, w) = (−D1 (D2 L(q, v)·) · v − D2 D2 L(q, v) · X2 (q, v) + D1 L(q, v)) · u. (7.8.8) ........................... 15 July 1998—18h02 ...........................

218. 204 7. Lagrangian Mechanics Thus, the local Lagrange–d’Alembert principle becomes (−D1 (D2 L(q, v)·) · v − D2 D2 L(q, v) · X2 (q, v) + D1 L(q, v) + D2 D2 L(q, v) · Y2 (q, v)) = 0. (7.8.9) Setting v = dq/dt and X2 (q, v) = dv/dt, the preceding relation and the chain rule gives d D2 L(q, v) − D1 L(q, v) = D2 D2 L(q, v) · Y2 (q, v), (7.8.10) dt which, in finite dimensions, reads d ∂L ∂L ∂2L − = i j Y j (q k , q k ). ˙ (7.8.11) dt ∂ qi ˙ ∂q i ∂q ∂q ˙ ˙ The force one-form ∆Y is therefore given by ∂2L ∆Y (q k , q k ) = ˙ Y j (q k , q k ) dq i ˙ (7.8.12) ∂ qi ∂ qj ˙ ˙ and the corresponding force field is ∂2L FY = qi , Y j (q k , q k ) . ˙ (7.8.13) ∂ qi ∂ qj ˙ ˙ Thus, the condition for an integral curve takes the form of the standard Euler–Lagrange equations with forces: d ∂L ∂L − = FiY (q k , q k ). ˙ (7.8.14) dt ∂ qi ˙ ∂q i Since the integral Lagrange–d’Alembert principle gives the same equations, it follows that the two principles are equivalent. From now on, we will refer to either one as simply the Lagrange–d’Alembert principle. We summarize the results obtained so far in the following: Theorem 7.8.6. Given a regular Lagrangian and a force field F : T Q → T ∗ Q, for a curve q(t) in Q, the following are equivalent: (a) q(t) satisfies the local Lagrange–d’Alembert principle; (b) q(t) satisfies the integral Lagrange–d’Alembert principle; and (c) q(t) is the base integral curve of the second-order equation Z + Y , where Y is the vertical vector field on T Q inducing the force field F by (7.8.13), and Z is the Lagrangian vector field on L. The Lagrange–d’Alembert principle plays a crucial role in nonholonomic mechanics, such as mechanical systems with rolling constraints. See, for example, Bloch, Krishnaprasad, Marsden, and Murray [1996] and references therein. ........................... 15 July 1998—18h02 ...........................

219. 7.8 The Lagrange–d’Alembert Principle 205 Dissipative Forces. Let E denote the energy defined by L, that is, E = A − L, where A(v) = FL(v), v is the action of L. Definition 7.8.7. A vertical vector field Y on T Q is called weakly dissipative if dE, Y ≤ 0 at all points of T Q. If the inequality is strict off the zero section of T Q, Y is called dissipative. A dissipative Lagrangian system on T Q is a vector field Z + Y , for Z a Lagrangian vector field and Y a dissipative vector field. Corollary 7.8.8. A vertical vector field Y on T Q is dissipative if and only if the force field F Y that it induces satisfies F Y (v), v 0 for all nonzero v ∈ T Q (≤ 0 for the weakly dissipative case). Proof. Let Y be a vertical vector field. By Proposition 7.8.1, Y induces a horizontal one-form ∆Y = −iY ΩL on T Q, and by Proposition 7.8.2 , ∆Y in turn induces a force field F Y given by F Y (v), w = ∆Y (v) · Vv = −ΩL (v)(Y (v), Vv ), (7.8.15) where T τQ (Vv ) = w and Vv ∈ Tv (T Q). If Z denotes the Lagrangian system defined by L, we get (dE · Y )(v) = (iZ ΩL )(Y )(v) = ΩL (Z, Y )(v) = −ΩL (v)(Y (v), Z(v)) = F Y (v), Tv τ (Z(v)) = F Y (v), v , since Z is a second-order equation. Thus, dE · Y 0 if and only if F Y (v), v 0 for all v ∈ T Q. Definition 7.8.9. Given a dissipative vector field Y on T Q, let F Y : T Q → T ∗ Q be the induced force field. If there is a function R : T Q → R such that F Y is the fiber derivative of −R, then R is called a Rayleigh dissipation function. Note that in this case, D2 R(q, v) · v 0 for the dissipativity of Y . Thus, if R is linear in the fiber variable, the Rayleigh dissipation function takes on the classical form R(q)v, v , where R(q) : T Q → T ∗ Q is a bundle map over the identity that defines a symmetric positive–definite form on each fiber of T Q. Finally, if the force field is given by a Rayleigh dissipation function R, then the Euler–Lagrange equations with forcing become d ∂L ∂L ∂R − = − i. (7.8.16) dt ∂ qi ˙ ∂q i ∂q ˙ ........................... 15 July 1998—18h02 ...........................

220. 206 7. Lagrangian Mechanics Combining Corollary 7.8.8 with the fact that the differential of E along Z is zero, we find that under the flow of the Euler–Lagrange equations with forcing of Rayleigh dissipation type d E(q, v) = F (v) · v = −FR(q, v) · v 0. (7.8.17) dt Exercises 7.8-1. What is the power or rate of work equation (see §2.1) for a system with forces on a Riemannian manifold? 7.8-2. Write the equations for a ball in a rotating hoop, including friction, in the language of this section. (See §2.10). Compute the Rayleigh dissipation function. 7.8-3. Consider a Riemannian manifold Q and a potential function V : Q → R. Let K denote the kinetic energy function and let ω = −dV . Show that the Lagrange–d’Alembert principle for K with external forces given by the one form ω produces the same dynamics as the standard kinetic minus potential Lagrangian. 7.9 The Hamilton–Jacobi Equation In §6.5 we studied generating functions of canonical transformations. Here we link them with the flow of a Hamiltonian system via the Hamilton– Jacobi equation. In this section we approach Hamilton–Jacobi theory from the point of view of extended phase space. In the next Chapter we will have another look at Hamilton–Jacobi theory from the variational point of view, as it was originally developed by Jacobi [1866]. In particular, we will show in that section, roughly speaking, that the integral of the Lagrangian along solutions of the Euler–Lagrange equations, but thought of as a function of the endpoints satisfies the Hamilton–Jacobi equation. Canonical Transformations and Generating Functions. We consider a symplectic manifold P and form the extended phase space P ×R. For our purposes in this section, we will use the following definition. A time dependent canonical transformation is a diffeomorphism ψ : P × R → P × R of the form ψ(z, t) = (ψt (z), t), where, for each t ∈ R, ψt : P → P is a symplectic diffeomorphism. We will also specialize in this section to cotangent bundles, so assume that P = T ∗ Q for a configuration manifold Q. For each fixed t, let St : Q × Q → R be a generating function for ψt as described in §6.5. Thus, we ........................... 15 July 1998—18h02 ...........................

221. 7.9 The Hamilton–Jacobi Equation 207 get a function S : Q × Q × R → R defined by S(q1 , q2 , t) = St (q1 , q2 ). As explained in §6.5, one has to be aware that, in general, generating functions are defined only locally and indeed the global theory of generating functions and the associated global Hamilton–Jacobi theory is more sophisticated. We will give a brief (optional) introduction to this general theory at the end of this section. See also Abraham and Marsden [1978, §5.3] for more information and references. Since our goal in the first part of this section is to give an introductory presentation of the theory, we will do many of the calculations in coordinates. Recall that in local coordinates, the conditions for a generating function are written as follows. If the transformation ψ has the local expression ψ : (q i , pi , t) → (q i , pi , t), and if S(q i , q i , t) is a generating function, we have the relations ∂S ∂S pi = − and pi = . (7.9.1) ∂q i ∂q i From (7.9.1) it follows that ∂S i ∂S i pi dq i = pi dq i + dq + i dq ∂q i ∂q ∂S = pi dq i − dt + dS, (7.9.2) ∂t where dS is the differential of S as a function on Q × Q × R: ∂S i ∂S i ∂S dS = dq + i dq + dt. ∂q i ∂q ∂t Let K : T ∗ Q × R → R be an arbitrary function. From (7.9.2), we get the following basic relationship: pi dq i − K(q i , pi , t) dt = pi dq i − K(q i , pi , t) dt + dS(q i , q i , t), (7.9.3) where K(q i , pi , t) = K(q i , pi , t) + ∂S(q i , q i , t)/∂t. If we define ΘK = pi dq i − K dt, (7.9.4) (7.9.3) is equivalent to ΘK = ψ ∗ ΘK + ψ ∗ dS, (7.9.5) where ψ : T ∗ Q × R → Q × Q × R is the map (q i , pi , t) → (q i , q i (q j , pj , t), t). ........................... 15 July 1998—18h02 ...........................

222. 208 7. Lagrangian Mechanics By taking the exterior derivative of (7.9.3) (or (7.9.5)), it follows that dq i ∧ dpi + dK ∧ dt = dq i ∧ dpi + dK ∧ dt. (7.9.6) This may be written as Ω K = ψ ∗ ΩK (7.9.7) where ΩK = −dΘK = dq i ∧ dpi + dK ∧ dt. Recall from Exercise 6.2-3 that given a time dependent function K, and associated time dependent vector field XK on T ∗ Q, the vector field XK = ˜ ∗ (XK , 1) on T Q × R is uniquely determined (amongst all vector fields with a one in the second component) by the equation iXK ΩK = 0. From this ˜ relation and (7.9.7), we get 0 = ψ∗ (iXK ΩK ) = iψ∗ (XK ) ψ∗ ΩK = iψ∗ (XK ) ΩK . ˜ ˜ ˜ Since ψ is the identity in the second component, that is, it preserves time, ˜ the vector field ψ∗ (XK ) has a one in the second component and therefore by uniqueness of such vector fields, we get the identity ˜ ˜ ψ∗ (XK ) = XK . (7.9.8) The Hamilton–Jacobi Equation. The data we shall need are a Hamil- tonian H and a generating function S, as above. Definition 7.9.1. Given a time dependent Hamiltonian H and a transformation ψ with generating function S as above, we say that the Hamilton– Jacobi equation holds if ∂S ∂S ∂S i i H q1 , . . . , qn , ,... , n,t + (q , q , t) = 0, (7.9.9) ∂q 1 ∂q ∂t in which ∂S/∂q i are evaluated at (q i , q i , t) and in which the q i are regarded as constants. The Hamilton–Jacobi equation may be regarded as a nonlinear partial differential equation for the function S relative to the variables (q 1 , . . . , q n , t) depending parametrically on (q 1 , . . . , q n ). Tudor, Jerry: Definition 7.9.2. We say that the map ψ transforms a vector field ˜ discuss how X to equilibrium if ψ is related ψ∗ X˜ = (0, 1) (7.9.10) to flow of X. ˜ If ψ transforms X to equilibrium, then the integral curves of X with ˜ initial conditions (q0 , p0 , t0 ) are given by i i (q i (t), pi (t), t) = ψ −1 (q i (q0 , p0 , t0 ), pi (q0 , p0 , t0 ), t + t0 ) i i i i (7.9.11) ........................... 15 July 1998—18h02 ...........................

223. 7.9 The Hamilton–Jacobi Equation 209 since the integral curves of the constant vector field (0, 1) are just straight ˜ lines in the t–direction and since ψ maps integral curves of X to those of (0, 1). In other words, if a map transforms a vector field X ˜ to equilibrium, ˜ the integral curves of X are represented by straight lines in the image space and so the vector field has been “integrated.” Theorem 7.9.3 (Hamilton–Jacobi). (i) Suppose that S satisfies the Hamilton–Jacobi equation for a given time dependent Hamiltonian H and that S generates a time dependent ˜ canonical transformation ψ. Then ψ transforms XH to equilibrium. Thus, as explained above, the solution of Hamilton’s equations for H are given in terms of ψ by (7.9.11). (ii) Conversely, if ψ is a time dependent canonical transformation with ˜ generating function S that transforms XH to equilibrium, then there ˆ that differs from S only by a function of t, which also is a function S, generates ψ, and satisfies the Hamilton–Jacobi equation for H. Proof. To prove (i), assume that S satisfies the Hamilton–Jacobi equation. As we explained above, this means that H = 0. From (7.9.8) we get ˜ ˜ ψ∗ XH = XH = (0, 1). This proves the first statement. To prove the converse (ii), assume that ˜ ψ∗ XH = (0, 1) and so, again by (7.9.8), ˜ ˜ XH = X0 = (0, 1) which means that H is a constant relative to the variables (q i , pi ) (its Hamiltonian vector field at each instant of time is zero) and thus, H = f (t), ˆ a function of time only. We can then modify S to S = S − F , where t F (t) = f (s)ds. This function, differing from S by a function of time alone, generates the same map ψ. Since ˆ 0 = H − f (t) = H + ∂S/∂t − dF/dt = H + ∂ S/∂t, ˆ ˆ and ∂S/∂q i = ∂ S/∂q i , we see that S satisfies the Hamilton–Jacobi equation for H. Remarks. 1. In general, the function S develops singularities or caustics as time increases, so it must be used with care. This process is, however, fundamental ........................... 15 July 1998—18h02 ...........................

224. 210 7. Lagrangian Mechanics in geometric optics and in quantization. Moreover, one has to be careful with the sense in which S generates the identity at t = 0 as it might have singular behavior in t. 2. Here is another link between the Lagrangian and Hamiltonian view of the Hamilton–Jacobi theory. Define S for t close to a fixed time t0 by the action integral t S(q i , q i , t) = ¯ L(q i (s), q i (s), s) ds, ˙ t0 where q i (s) is the solution of the Euler–Lagrange equation equalling q i at time t0 and equalling q i at time t. We will show in §8.2 that S satisfies the Hamilton–Jacobi equation. See Arnold [1989], §4.6, and Abraham and Marsden [1978], §5.2, for more information. 3. If H is time-independent and W satisfies the time-independent Ham- ilton–Jacobi equation ∂W H qi , = E, ∂q i then S(q i , q i , t) = W (q i , q i ) − tE satisfies the time-dependent Hamilton– ¯ Jacobi equation, as is easily checked. When using this remark, it is important to remember that E is not really a “constant”, but it equals H(q, p), the energy evaluated at (q, p), which will eventually be the initial conditions. We emphasize that one must generate the time t–map using S rather than W . 4. The Hamilton–Jacobi equation is fundamental in the study of the quantum-classical relationship is described in the optional §7.10. 5. The action function S is a key tool used in the proof of the Liouville– Arnold theorem which gives the existence of action angle coordinates for systems with integrals in involution; see Arnold [1989] and Abraham and Marsden [1978], for details. 6. The Hamilton–Jacobi equation plays an important role in the development of numerical integrators that preserve the symplectic structure (see deVogela´re [1956], Channell [1983], Feng [1986], Channell and Scovel e [1990], Ge and Marsden [1988], Marsden [1992], and Wendlandt and Mars- den [1997]). 7. The method of separation of variables. It is sometimes possible to simplify and even solve the Hamilton–Jacobi equation by what is often ........................... 15 July 1998—18h02 ...........................

225. 7.9 The Hamilton–Jacobi Equation 211 called the method of separation of variables. Assume that in the Hamilton– Jacobi equation the coordinate q 1 and the term ∂S/∂q 1 appear jointly in some expression f (q 1 , ∂S/∂q 1 ) that does not involve q 2 , . . . , q n , t. That is, we can write H in the form ˜ H q 1 , q 2 , . . . , q n , p1 , p2 , . . . , pn = H(f (q 1 , p1 ), q 2 , . . . , q n , p2 , . . . , pn ) ˜ for some smooth functions f and H. Then one seeks a solution of the Hamilton–Jacobi equation in the form ˜ S(q i , q i , t) = S1 (q 1 , q 1 ) + S(q 2 , . . . , q n , q 2 , . . . , q n ). We then note that if S1 solves ∂S1 f q1 , = C(q 1 ) ∂q 1 ˜ for an arbitrary function C(q 1 ) and if S solves ∂S˜ ∂S˜ ˜ ∂S ˜ H C(q 1 ), q 2 , . . . , q n , ,... , n + = 0, ∂q 2 ∂q ∂t then S solves the original Hamilton–Jacobi equation. In this way, one of the variables is eliminated and one tries to repeat the procedure. A closely related situation occurs when H is independent of time and one seeks a solution of the form S(q i , q i , t) = W (q i , q i ) + S1 (t). The resulting equation for S1 has the solution S1 (t) = −Et and the remaining equation for W is the time independent Hamilton–Jacobi equation as in Remark 3. If q 1 is a cyclic variable, that is, if H does not depend explicitly on q 1 , then we can choose f (q 1 , p1 ) = p1 and, correspondingly, we can choose S1 (q 1 ) = C(q 1 )q 1 . In general, if there are k cyclic coordinates q 1 , q 2 , . . . , q k we seek a solution to the Hamilton–Jacobi equation of the form k S(q i , q i , t) = ˜ Cj (q j )q j + S(q k+1 , . . . , q n , q k+1 , . . . , q n , t), j=1 with pi = Ci (q i ), i = 1, . . . , k being the momenta conjugate to the cyclic variables. Internet ........................... 15 July 1998—18h02 ........................... Supplement?

226. 212 7. Lagrangian Mechanics The Geometry of Hamilton–Jacobi Theory (Optional). Now, we describe briefly and informally, some additional geometry connected with ˜ the Hamilton–Jacobi equation (7.9.9). For each x = (q i , t) ∈ Q := Q × ∗˜ R, dS(x) is an element of the cotangent bundle T Q. We suppress the dependence of S on q i for the moment since it does not play an immediate ˜ ˜ role. As x varies in Q, the set {dS(x) | x ∈ Q} defines a submanifold of ∗˜ T Q which in terms of coordinates is given by pj = ∂S/∂q j and p = ∂S/∂t; here the variables conjugate to q i are denoted pi and that conjugate to t is denoted p. We will write ξi = pi for i = 1, 2, . . . , n and ξn+1 = p. We call this submanifold the range, or graph of dS (either term is appropriate, depending on whether one thinks of dS as a mapping or as a section of a bundle) and denote it by graph dS ⊂ T ∗ Q. The restriction of the canonical ˜ ∗˜ symplectic form on T Q to graph dS is zero since n+1 n+1 n+1 ∂S ∂2S dx ∧ dξj = j dx ∧ d j = dxj ∧ dxk = 0. j=1 j=1 ∂xj ∂xj ∂xk j,k=1 Moreover, the dimension of the submanifold graph dS is half of the dimension of the symplectic manifold T ∗ Q. Such a submanifold is called ˜ Lagrangian, as we already mentioned in connection with generating functions (§6.5). What is important here is that the projection from graph dS to Q is a diffeomorphism, and even more, the converse holds: if Λ ⊂ T ∗ Q is ˜ ˜ ∗˜ ˜ is a diffeo- a Lagrangian submanifold of T Q such that the projection on Q morphism in a neighborhood of a point λ ∈ Λ, then in some neighborhood of λ, we can write Λ = graph dϕ for some function ϕ. To show this, notice that because the projection is a diffeomorphism, Λ is given (around λ) as a submanifold of the form (xj , ρj (x)). The condition for Λ to be Lagrangian requires that, on Λ, n+1 dxj ∧ dξj = 0 j=1 that is, n+1 ∂ρj ∂ρk dxj ∧ dρj (x) = 0, i.e., k − = 0; j=1 ∂x ∂xj thus, there is a ϕ such that ρj = ∂ϕ/∂xj , which is the same as Λ = graph dϕ. The conclusion of these remarks is that Lagrangian submanifolds of T ∗ Q are natural generalizations of graphs of differentials of functions on ˜ ˜ Q. Note that Lagrangian submanifolds are defined even if the projection ˜ to Q is not a diffeomorphism. For more information on Lagrangian manifolds and generating functions, see Abraham and Marsden [1978], Weinstein [1977] and Guillemin and Sternberg [1977]. ........................... 15 July 1998—18h02 ...........................

227. 7.9 The Hamilton–Jacobi Equation 213 From the point of view of Lagrangian submanifolds, the graph of the differential of a solution of the Hamilton–Jacobi equation is a Lagrangian submanifold of T ∗ Q which is contained in the surface H0 ⊂ T ∗ Q defined ˜ ˜ ˜ by the equation H ˜ := p + H(q i , pi , t) = 0. Here, as above, p = ξn+1 is the momentum conjugate to t. This point of view allows one to include solutions which are singular in the usual context. This is not the only benefit: we also get more insight in the content of the Hamilton–Jacobi ˜ Theorem 7.9.3. The tangent space to H0 has dimension 1 less than the dimension of the symplectic manifold T ∗ Q and it is given by the set of ˜ vectors X such that (dp + dH)(X) = 0. If a vector Y is in the symplectic ˜ orthogonal of T(x,ξ) (H0 ), that is, n+1 (dxj ∧ dξj )(X, Y ) = 0 j=1 ˜ for all X ∈ T(x,ξ) (H0 ), then Y is a multiple of the vector field ∂ ∂H ∂ XH = ˜ − + XH ∂t ∂t ∂p evaluated at (x, ξ). Moreover, the integral curves of XH projected to (q i , pi ) ˜ are the solutions of Hamilton’s equations for H. The key observation that links Hamilton’s equations and the Hamilton– Jacobi equation is that the vector field XH which is obviously tangent to ˜ ˜ H0 is, moreover, tangent to any Lagrangian submanifold contained in H0 ˜ (the reason for this is a very simple algebraic fact given in Exercise 7.9- 3). This is the same as saying that a solution of Hamilton’s equations for ˜ H is either disjoint from a Lagrangian submanifold contained in H0 or ˜ completely contained in it. This gives a way to construct a solution of the Hamilton–Jacobi equation starting from an initial condition at t = t0 . Namely, take a Lagrangian submanifold Λ0 in T ∗ Q and embed it in T ∗ Q ˜ at t = t0 using (q i , pi ) → (q i , t = t0 , pi , p = −H(q i , pi , t0 )). The result is an isotropic submanifold Λ0 ⊂ T ∗ Q; that is, a submanifold ˜ ˜ on which the canonical form vanishes. Now take all integral curves of XH ˜ ˜ whose initial conditions lie in Λ0 . The collection of these curves spans a ˜ manifold Λ whose dimension is one higher than Λ0 . It is obtained by flowing ˜ 0 along X ˜ ; that is, Λ = ∪t Λt , where Λt = Φt (Λ0 ) and Φt is the flow of Λ ˜ H XH . Since XH is tangent to H ˜ ˜ ˜ 0 and Λ0 ⊂ H0 , we get Λt ⊂ H0 and hence ˜ ˜ ˜ Λ ⊂ H0 . Since the flow Φt of XH is a canonical map, it leaves the symplectic ˜ form of T ∗ Q invariant and therefore takes an isotropic submanifold into an ˜ isotropic one; in particular Λt is an isotropic submanifold of T ∗ Q. The ˜ tangent space of Λ at some λ ∈ Λt is a direct sum of the tangent space of ........................... 15 July 1998—18h02 ...........................

228. 214 7. Lagrangian Mechanics Λt and the subspace generated by XH ; since the first subspace is contained ˜ ˜ ˜ in Tλ H0 and the second is symplectically orthogonal to Tλ H0 , we see that Λ is also an isotropic submanifold of T ∗ Q. But its dimension is half that ˜ of T ∗ Q and therefore Λ is a Lagrangian submanifold contained in H0 , that ˜ ˜ is, it is a solution of the Hamilton–Jacobi equation with initial condition Λ0 at t = t0 . Using the above point of view it is easy to understand the singularities of a solution of Hamilton–Jacobi equation. They correspond to those points ˜ of the Lagrangian manifold solution where the projection to Q is not a local diffeomorphism. These singularities might be present in the initial condition (that is, Λ0 might not locally project diffeomorphically to Q) or they might appear at later times by folding the submanifolds Λt as t varies. ˜ The projection of such a singular point to Q is called a caustic point of the solution. Caustic points are of fundamental importance in geometric optics and the semiclassical approximation of quantum mechanics. We refer to Abraham and Marsden [1978] §5.3 and Guillemin and Sternberg [1984] for further information. Exercises 7.9-1. Solve the Hamilton–Jacobi equation for the harmonic oscillator. Check directly the validity of the Hamilton–Jacobi theorem (connecting the solution of the Hamilton–Jacobi equation and the flow of the Hamiltonian vector field) for this case. 7.9-2. Verify by direct calculation the following. Let W (q, q) and p2 H(q, p) = + V (q) 2m be given, where q, p ∈ R. Show that for p = 0, 1 (Wq )2 + V = E 2m and q = p/m if and only if (q, Wq (q, q)) satisfies Hamilton’s equation with ˙ energy E. 7.9-3. Let (V, Ω) be a symplectic vector space and W ⊂ V be a linear subspace. Recall from §2.4 that W Ω = {v ∈ V | Ω(v, w) = 0 for all w ∈ W } denotes the symplectic orthogonal of W . A subspace L ⊂ V is called Lagrangian if L = LΩ . Show that if L ⊂ W is a Lagrangian subspace, then W Ω ⊂ L. 7.9-4. Solve the Hamilton–Jacobi equation for a central force field. Check directly the validity of the Hamilton–Jacobi theorem. ........................... 15 July 1998—18h02 ...........................

229. This is page 215 Printer: Opaque this 8 Variational Principles, Constraints, and Rotating Systems This chapter deals with two related topics: constrained Lagrangian (and Hamiltonian) systems and rotating systems. Constrained systems are illus- trated by a particle constrained to move on a sphere. Such constraints that involve conditions on the configuration variables are called “holonomic.”1 For rotating systems, one needs to distinguish systems that are viewed from rotating coordinate systems (passively rotating systems) and systems which themselves are rotated (actively rotating systems—such as a Fou- cault pendulum and weather systems rotating with the Earth). We begin with a more detailed look at variational principles and then we turn to a version of the Lagrange multiplier theorem that will be useful for our analysis of constraints. 8.1 A Return to Variational Principles In this section we take a closer look at variational principles. Technicalities involving infinite-dimensional manifolds prevent us from presenting the full story from that point of view. For these, we refer to, for example, Smale [1964], Palais [1968], and Klingenberg [1978]. For the classical geometric theory without the infinite-dimensional framework, the reader may consult, 1 In this volume we shall not discuss “nonholonomic” constraints such as rolling constraints. We refer to Bloch, Krishnaprasad, Marsden, and Murray [1996], and Bloch et al. [1998] for a discussion of nonholonomic systems and further refrerences.

230. 216 8. Variational Principles, Constraints, and Rotating Systems for example, Bolza [1973], Whittaker [1927], Gelfand and Fomin [1963], or Hermann [1968]. Hamilton’s Principle. We begin by setting up the space of paths joining two points. Definition 8.1.1. Let Q be a manifold and let L : T Q → R be a regular Lagrangian. Fix two points q1 and q2 in Q and an interval [a, b], define the path space from q1 to q2 by Ω(q1 , q2 , [a, b]) = {c : [a, b] → Q | c is a C 2 curve, c(a) = q1 , c(b) = q2 }, (8.1.1) and the map S : Ω(q1 , q2 , [a, b]) → R by b S(c) = L(c(t), c(t)) dt. ˙ a What we shall not prove is that Ω(q1 , q2 , [a, b]) is a smooth infinite-dimensional manifold. This is a special case of a general result in the topic of manifolds of mappings, wherein spaces of maps from one manifold to another are shown to be smooth infinite-dimensional manifolds. Accepting this, we can prove the following. Proposition 8.1.2. The tangent space, Tc Ω(q1 , q2 , [a, b]), to the manifold Ω(q1 , q2 , [a, b]) at a point, that is, a curve c ∈ Ω(q1 , q2 , [a, b]), is the set of C 2 maps v : [a, b] → T Q such that τQ ◦ v = c and v(a) = 0, v(b) = 0, where τQ : T Q → Q denotes the canonical projection. Proof. The tangent space to a manifold consists of tangents to smooth curves in the manifold. The tangent vector to a curve cλ ∈ Ω(q1 , q2 , [a, b]) with c0 = c is d v= cλ . (8.1.2) dλ λ=0 However, cλ (t), for each fixed t, is a curve through c0 (t) = c(t). Hence d cλ (t) dλ λ=0 is a tangent vector to Q based at c(t). Hence v(t) ∈ Tc(t) Q; that is, τQ ◦v = c. The restrictions cλ (a) = q1 and cλ (b) = q2 lead to v(a) = 0 and v(b) = 0, but otherwise v is an arbitrary C 2 function. One refers to v as an infinitesimal variation of the curve c subject to fixed endpoints and we use the notation v = δc. See Figure 8.1.1. Now we can state and sketch the proof of a main result in the calculus of variations in a form due to Hamilton [1830]. ........................... 15 July 1998—18h02 ...........................

231. 8.1 A Return to Variational Principles 217 q(t) δq(t) q(b) q(a) Figure 8.1.1. The variation δq(t) of a curve q(t) is a field of vectors along that curve. Theorem 8.1.3 (Variational Principle of Hamilton). Let L be a La- grangian on T Q. A curve c0 : [a, b] → Q joining q1 = c0 (a) to q2 = c0 (b) satisfies the Euler–Lagrange equations d ∂L ∂L = , (8.1.3) dt ∂ qi ˙ ∂q i if and only if c0 is a critical point of the function S : Ω(q1 , q2 , [a, b]) → R, that is, dS(c0 ) = 0. If L is regular, either condition is equivalent to c0 being a base integral curve of XE . As in §7.1, the condition dS(c0 ) = 0 is denoted b δ L(c0 (t), c0 (t)) dt = 0; ˙ (8.1.4) a that is, the integral is stationary when it is differentiated with c regarded as the independent variable. Proof. We work out dS(c) · v just as in §7.1. Write v as the tangent to the curve cλ in Ω(q1 , q2 , [a, b]) as in (8.1.2). By the chain rule, b d d dS(c) · v = S(cλ ) = L(cλ (t), cλ (t)) dt ˙ . (8.1.5) dλ λ=0 dλ a λ=0 Differentiating (8.1.5) under the integral sign, and using local coordinates,2 we get b ∂L i ∂L i dS(c) · v = v + i v dt. ˙ (8.1.6) a ∂q i ∂q ˙ 2 If the curve c (t) does not lie in a single coordinate chart, divide the curve c(t) into 0 a finite partition each of whose elements lies in a chart and apply the argument below. ........................... 15 July 1998—18h02 ...........................

232. 218 8. Variational Principles, Constraints, and Rotating Systems Since v vanishes at both ends, the second term in (8.1.6) can be integrated by parts to give b ∂L d ∂L dS(c) · v = i − v i dt. (8.1.7) a ∂q dt ∂ q i ˙ Now dS(c) = 0 means dS(c) · v = 0 for all v ∈ Tc Ω(q1 , q2 , [a, b]). This holds if and only if ∂L d ∂L i − = 0, (8.1.8) ∂q dt ∂ qi ˙ since the integrand is continuous and v is arbitrary, except for v = 0 at the ends. (This last assertion was proved in Theorem 7.3.3.) The reader can check that Hamilton’s principle proceeds virtually unchanged for time-dependent Lagrangians. We shall use this remark below. The Principle of Critical Action. Next we discuss variational principles with the constraint of constant energy imposed. To compensate for this constraint, we let the interval [a, b] be variable. Deﬁnition 8.1.4. Let L be a regular Lagrangian and let Σe be a regular energy surface for the energy E of L, that is, e is a regular value of E and Σe = E −1 (e). Let q1 , q2 ∈ Q and let [a, b] be a given interval. Deﬁne Ω(q1 , q2 , [a, b], e) to be the set of pairs (τ, c), where τ : [a, b] → R is C 2 , τ ˙ 0, and where c : [τ (a), τ (b)] → Q is a C 2 curve with c(τ (a)) = q1 , c(τ (b)) = q2 , and E (c(τ (t)), c(τ (t))) = e, ˙ for all t ∈ [a, b]. Arguing as in Proposition 8.1.2, computation of the derivatives of curves (τλ , cλ ) in Ω(q1 , q2 , [a, b], e) shows that the tangent space to Ω(q1 , q2 , [a, b], e) at (τ, c) consists of the space of pairs of C 2 maps α : [a, b] → R and v : [τ (a), τ (b)] → T Q such that v(t) ∈ Tc(t) Q, c(τ (a))α(a) + v(τ (a)) = 0 ˙ (8.1.9) c(τ (b))α(b) + v(τ (b)) = 0 ˙ and dE[c(τ (t)), c(τ (t))] · [c(τ (t))α(t) + v(τ (t)), c(τ (t))α(t) + v(τ (t))] = 0. ˙ ˙ ¨ ˙ ˙ (8.1.10) ........................... 15 July 1998—18h02 ...........................

233. 8.1 A Return to Variational Principles 219 Theorem 8.1.5 (Principle of Critical Action). Let c0 (t) be a solution of the Euler–Lagrange equations and let q1 = c0 (a) and q2 = c0 (b). Let e be the energy of c0 (t) and assume it is a regular value of E. Define the map A : Ω(q1 , q2 , [a, b], e) → R by τ (b) A(τ, c) = A(c(t), c(t)) dt, ˙ (8.1.11) τ (a) where A is the action of L. Then dA(Id, c0 ) = 0, (8.1.12) where Id is the identity map. Conversely, if (Id, c0 ) is a critical point of A and c0 has energy e, a regular value of E, then c0 is a solution of the Euler–Lagrange equations. In coordinates, (8.1.11) reads τ (b) τ (b) ∂L i A(τ, c) = q dt = ˙ pi dq i , (8.1.13) τ (a) ∂ qi ˙ τ (a) the integral of the canonical one-form along the curve γ = (c, c). Being the ˙ line integral of a one-form, A(τ, c) is independent of the parametrization τ . Thus, one may think of A as defined on the space of (unparametrized) curves joining q1 and q2 . Proof. If the curve c has energy e, then τ (b) A(τ, c) = [L(q i , q i ) + e] dt. ˙ τ (a) Differentiating A with respect to τ and c by the method of Theorem 8.1.3 gives dA(Id, c0 ) · (α, v) = α(b) [L(c0 (b), c0 (b)) + e] − α(a) [L(c0 (a), c0 (a)) + e] ˙ ˙ b ∂L ∂L + (c (t), c0 (t))v i (t) + i (c0 (t), c0 (t))v i (t) dt. i 0 ˙ ˙ ˙ a ∂q ∂q ˙ (8.1.14) Integrating by parts gives dA(Id, c0 ) · (α, v) b ∂L = α(t) [L(c0 (t), c0 (t)) + e] + ˙ (c0 (t), c0 (t))v i (t) ˙ ∂ qi ˙ a b ∂L d ∂L + (c (t), c0 (t)) − i 0 ˙ (c0 (t), c0 (t)) v i (t) dt. (8.1.15) ˙ a ∂q dt ∂ q i ˙ ........................... 15 July 1998—18h02 ...........................

234. 220 8. Variational Principles, Constraints, and Rotating Systems Using the boundary conditions v = −cα, noted in the description of the tan- ˙ gent space T(Id,c0 ) Ω(q1 , q2 , [a, b], e) and the energy constraint (∂L/∂ q i )ci − ˙ ˙ L = e, the boundary terms cancel, leaving b ∂L d ∂L dA(Id, c0 ) · (α, v) = − v i dt. (8.1.16) a ∂q i dt ∂ q i ˙ However, we can choose v arbitrarily; notice that the presence of α in the linearized energy constraint means that no restrictions are placed on the variations v i on the open set where c = 0. The result therefore follows. ˙ If L = K −V , where K is the kinetic energy of a Riemannian metric, then Theorem 8.1.5 states that a curve c0 is a solution of the Euler–Lagrange equations if and only if b δe 2K(c0 , c0 ) dt = 0, ˙ (8.1.17) a where δe indicates a variation holding the energy and endpoints but not the parametrization fixed; this is symbolic notation for the precise statement in Theorem 8.1.5. Using the fact that K ≥ 0, a calculation of the Euler– Lagrange equations (Exercise 8.1-3) shows that (8.1.17) is the same as b δe 2K(c0 , c0 ) dt = 0, ˙ (8.1.18) a that is, arc length is extremized (subject to constant energy). This is Ja- cobi’s form of the principle of “least action” and represents a key to linking mechanics and geometric optics, which was one of Hamilton’s original motivations. In particular, geodesics are characterized as extremals of arc length. Using the Jacobi metric (see §7.7) one gets yet another variational principle.3 Phase Space Form of the Variational Principle. The above variational principles for Lagrangian systems carry over to some extent to Hamiltonian systems. Theorem 8.1.6 (Hamilton’s Principle in Phase Space). Consider a Hamiltonian H on a given cotangent bundle T ∗ Q. A curve (q i (t), pi (t)) in T ∗ Q satisfies Hamilton’s equations iff b δ [pi q i − H(q i , pi )] dt = 0 ˙ (8.1.19) a for variations over curves (q i (t), pi (t)) in phase space, where q i = dq i /dt ˙ and where q i are fixed at the endpoints. 3 Other interesting variational principles are those of Gauss, Hertz, Gibbs, and Appell. A modern account, along with references, is Lewis [1997] ........................... 15 July 1998—18h02 ...........................

235. 8.1 A Return to Variational Principles 221 Proof. Computing as in (8.1.6), we find that b b ∂H i ∂H δ [pi q i − H(q i , pi )] dt = ˙ (δpi )q i + pi (δ q i ) − ˙ ˙ δq − δpi dt. a a ∂q i ∂pi (8.1.20) Since q i (t) are fixed at the two ends, we have pi δq i = 0 at the two ends, and hence the second term of (8.1.20) can be integrated by parts to give b ∂H i ∂H q i (δpi ) − pi (δq i ) − ˙ ˙ δq − δpi dt, (8.1.21) a ∂q i ∂pi which vanishes for all δpi , δq i exactly when Hamilton’s equations hold. Hamilton’s principle in phase space (8.1.19) on an exact symplectic manifold (P, Ω = −dΘ) reads b δ (Θ − Hdt) = 0, (8.1.22) a again with suitable boundary conditions. Likewise, if we impose the constraint H = constant, the principle of least action reads τ (b) δ Θ = 0. (8.1.23) τ (a) In Cendra and Marsden [1987], Cendra, Ibort, and Marsden [1987], and Marsden and Scheurle [1993a,b], it is shown how to form variational principles on certain symplectic and Poisson manifolds even when Ω is not exact, but does arise by a reduction process. The variational principle for the Euler–Poincar´ equations that was described in the introduction and e that we shall encounter again in Chapter 13, is a special instance of this. The one-form ΘH := Θ − Hdt in (8.1.22), regarded as a one-form on P × R is an example of a contact form and plays an important role in time-dependent and relativistic mechanics. Let ΩH = −dΘH = Ω + dH ∧ dt and observe that the vector field XH is characterized by the statement that ˜ its suspension XH = (XH , 1), a vector field on P × R, lies in the kernel of ΩH : iXH ΩH = 0. ˜ ........................... 15 July 1998—18h02 ...........................

236. 222 8. Variational Principles, Constraints, and Rotating Systems Exercises 8.1-1. In Hamilton’s principle, show that the boundary conditions of fixed q(a) and q(b) can be changed to p(b) · δq(b) = p(a) · δq(a). What is the corresponding statement for Hamilton’s principle in phase space? 8.1-2. Show that the equations for a particle in a magnetic field B and a potential V can be written as e δ (K − V ) dt = − δq · (v × B) dt. c 8.1-3. Do the calculation showing that b δe 2K(c0 , c0 ) dt = 0, ˙ a and b δe 2K(c0 , c0 ) dt = 0, ˙ a are equivalent. 8.2 The Geometry of Variational Principles In Chapter 7 we derived the “geometry” of Lagrangian systems on T Q by pulling back the geometry from the Hamiltonian side on T ∗ Q. Now we show how all of this basic geometry of Lagrangian systems can be derived directly from Hamilton’s principle. The exposition below follows Marsden, Patrick, and Shkoller [1998]. A Brief Review. Recall that given a Lagrangian function L : T Q → R, we construct the corresponding action functional S on C 2 curves q(t), a ≤ t ≤ b by (using coordinate notation) b dq i S q(·) ≡ L q i (t), (t) dt. (8.2.1) a dt Hamilton’s principle (Theorem 8.1.3) seeks the curves q(t) for which the functional S is stationary under variations of q i (t) with fixed endpoints at fixed times. Recall that this calculation gives b b ∂L d ∂L ∂L i dS q(·) · δq(·) = δq i i − dt + δq . (8.2.2) a ∂q dt ∂ q i ˙ ∂ qi ˙ a The last term in (8.2.2) vanishes since δq(a) = δq(b) = 0, so that the requirement that q(t) be stationary for S yields the Euler–Lagrange equations ∂L d ∂L − = 0. (8.2.3) ∂q i dt ∂ q i ˙ ........................... 15 July 1998—18h02 ...........................

237. 8.2 The Geometry of Variational Principles 223 Recall that L is called regular when the matrix [∂ 2 L/∂ q i ∂ q j ] is everywhere ˙ ˙ a nonsingular matrix and in this case, the Euler–Lagrange equations are second-order ordinary differential equations for the required curves. Since the action (8.2.1) is independent of the choice of coordinates, the Euler–Lagrange equations are coordinate independent as well. Conse- quently, it is natural that the Euler–Lagrange equations may be intrinsically expressed using the language of differential geometry. Recall that one defines the canonical 1-form Θ on the 2n-dimensional cotangent bundle T ∗ Q of Q by Θ(αq ) · wαq = αq , Tαq πQ (wαq ) , where αq ∈ Tq Q, wαq ∈ Tαq T ∗ Q, and πQ : T ∗ Q → Q is the projection. ∗ The Lagrangian L defines a fiber preserving bundle map FL : T Q → T ∗ Q, the Legendre transformation, by fiber differentiation: d FL(vq ) · wq = L(vq + wq ). d =0 One normally defines the Lagrange 1-form on T Q by pull-back, ΘL = FL∗ Θ, and the Lagrange 2-form by ΩL = −dΘL . We then seek a vector field XE (called the Lagrange vector field) on T Q such that XE ΩL = dE, where the energy E is defined by E(vq ) = FL(vq ), vq − L(vq ) = ΘL (XE )(vq ) − L(vq ). If FL is a local diffeomorphism, which is equivalent to L being regular, then XE exists and is unique, and its integral curves solve the Euler– Lagrange equations. The Euler–Lagrange equations are second-order equations in T Q. In addition, the flow Ft of XE is symplectic; that is, preserves ΩL : Ft∗ ΩL = ΩL . These facts were proved using differential forms and Lie derivatives in the last three chapters. The Variational Approach. Besides being more faithful to history, sometimes there are advantages to staying on the “Lagrangian side”. Many examples can be given, but the theory of Lagrangian reduction (the Euler- Poincar´ equations being an instance) is one example. Other examples are e the direct variational approach to questions in black hole dynamics given by Wald [1993] and the development of variational asymptotics (see Holm [1996], Holm, Marsden, and Ratiu [1998b], and references therein). In such studies, it is the variational principle that is the center of attention. The development begins by removing the endpoint condition δq(a) = δq(b) = 0 from (8.2.2) but still keeping the time interval fixed. Equa- tion (8.2.2) becomes b b ∂L d ∂L ∂L i dS q(·) · δq(·) = δq i i − dt + δq , (8.2.4) a ∂q dt ∂ q i ˙ ∂ qi ˙ a ........................... 15 July 1998—18h02 ...........................

238. 224 8. Variational Principles, Constraints, and Rotating Systems but now the left side operates on more general δq and, correspondingly, the last term on the right side need not vanish. That last term of (8.2.4) is a linear pairing of the function ∂L/∂ q i , a function of q i and q i , with the ˙ ˙ tangent vector δq i . Thus, one may consider it a 1-form on T Q; namely the Lagrange 1-form (∂L/∂ q i )dq i . ˙ Theorem 8.2.1. Given a C k Lagrangian L, k ≥ 2, there exists a unique C k−2 mapping DEL L : Q → T ∗ Q, defined on the second-order submanifold ¨ ¨ d2 q Q := (0) ∈ T (T Q) q is a C 2 curve in Q dt2 of T (T Q), and a unique C k−1 1-form ΘL on T Q, such that, for all C 2 variations q (t) (on a fixed t-interval ) of q(t), where q0 (t) = q(t), we have b b d2 q dq ˆ dS q(·) · δq(·) = DEL L · δq dt + ΘL · δq , (8.2.5) a dt2 dt a where d ˆ d d δq(t) = q (t), δq(t) = q (t). d =0 d =0 dt The 1-form so defined is a called the Lagrange 1-form. Indeed, uniqueness and local existence follow from the calculation (8.2.2). The coordinate independence of the action implies the global existence of DEL and the 1-form ΘL . Thus, using the variational principle, the Lagrange 1-form ΘL is the “boundary part” of the the functional derivative of the action when the boundary is varied. The analogue of the symplectic form is the negative exterior derivative of ΘL ; that is, ΩL ≡ −dΘL . Lagrangian Flows are Symplectic. One of Lagrange’s basic discover- ies was that the solutions of the Euler–Lagrange equations give rise to a symplectic map. It is a curious twist of history that he did this without the machinery of either differential forms, or the Hamiltonian formalism, or of Hamilton’s principle itself. Assuming that L is regular, the variational principle gives coordinate independent second-order ordinary differential equations. We temporarily denote the vector field on T Q so obtained by X, and its flow by Ft . Now consider the restriction of S to the subspace CL of solutions of the variational principle. The space CL may be identified with the initial conditions for the flow; to vq ∈ T Q, we associate the integral curve s → Fs (vq ), s ∈ [0, t]. The value of S on the base integral curve q(s) = πQ (Fs (vq )) is denoted by St , that is, t St = L(Fs (vq )) ds, (8.2.6) 0 ........................... 15 July 1998—18h02 ...........................

239. 8.2 The Geometry of Variational Principles 225 and again called the action. We regard St as a real valued function on T Q. Note that by (8.2.6), dSt /dt = L(Ft (vq )). The fundamental equation (8.2.5) becomes d dSt (vq ) · wvq = ΘL Ft (vq ) · Ft (vq + wvq ) − ΘL (vq ) · wvq , d =0 where → vq + wvq symbolically represents a curve at vq in T Q with derivative wvq . Note that the first term on the right-hand side of (8.2.5) vanishes since we have restricted S to solutions. The second term becomes the one stated, remembering that now St is regarded as a function on T Q. We have thus derived the equation dSt = Ft∗ ΘL − ΘL . (8.2.7) Taking the exterior derivative of (8.2.7) yields the fundamental fact that the flow of X is symplectic: 0 = ddSt = d(Ft∗ ΘL − ΘL ) = −Ft∗ ΩL + ΩL which is equivalent to Ft∗ ΩL = ΩL . Thus, using the variational principle, the analogue that the evolution is symplectic is the equation d2 = 0, applied to the action restricted to the space of solutions of the variational principle. Equation (8.2.7) also provides the differential-geometric equations for X. Indeed, taking one time-derivative of (8.2.7) gives dL = £X ΘL , so that X ΩL = −X dΘL = −£X ΘL + d(X ΘL ) = d(X ΘL − L) = dE, where we define E = X ΘL − L. Thus, quite naturally, we find that X = XE . The Hamilton–Jacobi Equation. Next, we give a derivation of the Hamilton–Jacobi equation from variational principles. Allowing L to be time-dependent, Jacobi [1866] showed that the action integral defined by t S(q i , q i , t) = L(q i (s), q i (s), s) ds, ˙ t0 i where q (s) is the solution of the Euler–Lagrange equation subject to the conditions q i (t0 ) = q i and q i (t) = q i , satisfies the Hamilton–Jacobi equation. There are several implicit assumptions in Jacobi’s argument: L is regular and the time |t − t0 | is assumed to be small so that by the convex neighborhood theorem, S is a well defined function of the endpoints. We can allow |t − t0 | to be large as long as the solution q(t) is near a nonconjugate solution. Theorem 8.2.2 (Hamilton–Jacobi). With the above assumptions, the function S(q, q, t) satisfies the Hamilton–Jacobi equation: ∂S ∂S + H q, ,t = 0. ∂t ∂q ........................... 15 July 1998—18h02 ...........................

240. 226 8. Variational Principles, Constraints, and Rotating Systems Proof. In this equation, q is held fixed. Define v, a tangent vector at q, implicitly by πQ Ft (v) = q, (8.2.8) where Ft : T Q → T Q is the flow of the Euler–Lagrange equations, as in Theorem 7.4.5. As before, identifying the space of solutions CL of the Euler–Lagrange equations with the set of initial conditions, which is T Q, we regard t St (vq ) := S(q, q, t) := L(Fs (vq ), s) ds (8.2.9) 0 as a real-valued funcion on T Q. Thus, by the chain rule, and our previous calculations for St (see (8.2.7)), equation (8.2.9) gives ∂S ∂St ∂v = + dSt · ∂t ∂t ∂t ∂v ∂v = L(Ft (v), t) + (Ft∗ ΘL ) − ΘL , (8.2.10) ∂t ∂t where ∂v/∂t is computed by keeping q and q fixed and only changing t. Notice that in (8.2.10), q and q are held fixed on both sides of the equation; ∂S/∂t is a partial and not a total time-derivative. Implicitly differentiating the defining condition (8.2.8) with respect to t gives ∂v T πQ · XE (Ft (v)) + T πQ · T Ft · =0 ∂t Thus, since T πQ · XE (u) = u by the second-order equation property, we get ∂v T πQ · T Ft · = −q,˙ ∂t where (q, q) = Ft (v) ∈ Tq Q. Thus, ˙ ∂v ∂L i (Ft∗ ΘL ) = q. ˙ ∂t ∂ qi ˙ Also, since the base point of v does not change with t, T πQ · (∂v/∂t) = 0, so ΘL (∂v/∂t) = 0. Thus, (8.2.10) becomes ∂S ∂L = L(q, q, t) − ˙ q = −H(q, p, t). ˙ ∂t ∂q ˙ where p = ∂L/∂ q as usual. ˙ It remains only to show that ∂S/∂q = p. To do this, we diferentiate (8.2.8) implicitly with respect to q to give T πQ · T Ft (v) · (Tq v · u) = u. (8.2.11) ........................... 15 July 1998—18h02 ...........................

241. 8.2 The Geometry of Variational Principles 227 Then, from (8.2.9) and (8.2.7), Tq S(q, q, t) · u = dSt (v) · (Tq v · u) = (Ft∗ ΘL ) (Tq v · u) − ΘL (Tq v · u). As in (8.2.10), the last term vanishes since the base point q of v is fixed. Then, letting p = FL(Ft (v)), we get, from the definition of ΘL and pull- back, (Ft∗ ΘL ) (Tq v · u) = p, T πQ · T Ft (v) · (Tq v · u) = p, u in view of (8.2.11). The fact that ∂S/∂q = p also follows from the definition of S and the fundamental formula (8.2.4) . Just as we derived p = ∂S/∂q, we can derive ∂S/∂q = −p; in other words, S is the generating function for the canonical transformation (q, p) → (q, p). Some History of the Euler–Lagrange Equations. In the following paragraphs we make a few historical remarks concerning the Euler– Lagrange equations.4 Naturally, much of the story focuses on Lagrange. Section V of Lagrange’s Mćanique Analytique [1788] contains the equa- e tions of motion in Euler–Lagrange form (8.1.3). Lagrange writes Z = T −V for what we would call the Lagrangian today. In the previous section La- grange came to these equations by asking for a coordinate invariant expression for mass times acceleration. His conclusion is that it is given (in abbreviated notation) by (d/dt)(∂T /∂v) − ∂T /∂q, which transforms under arbitrary substitutions of position variables as a one-form. Lagrange does not recognize the equations of motion as being equivalent to the variational principle δ L dt = 0 —this was observed only a few decades later by Hamilton [1830]. The pecu- liar fact about this is that Lagrange did know the general form of the differential equations for variational problems and he actually had commented on Euler’s proof of this—his early work on this in 1759 was admired very much by Euler. He immediately applied it to give a proof of the Maupertuis principle of least action, as a consequence of Newton’s equations of motion. This principle, apparently having its roots in the early work of Leibniz, is 4 Many of these interesting historical points were conveyed to us by Hans Duistermaat to whom we are very grateful. The reader can also profitably consult some of the standard texts such as those of Whittaker [1927], Wintner [1941], and Lanczos [1949] for additional interesting historical information. ........................... 15 July 1998—18h02 ...........................

242. 228 8. Variational Principles, Constraints, and Rotating Systems a less natural principle in the sense that the curves are only varied over those which have a constant energy. It is also Hamilton’s principle that applies in the time-dependent case, when H is not conserved and which also generalizes to allow for certain external forces as well. This discussion in the Mćanique Analytique precedes the equations of e motion in general coordinates, and so is written in the case that the kinetic 2 energy is of the form i mi vi , where the mi are positive constants. Wintner [1941] is also amazed by the fact that the more complicated Maupertuis principle precedes Hamilton’s principle. One possible explanation is that Lagrange did not consider L as an interesting physical quantity—for him it was only a convenient function for writing down the equations of motion in a coordinate-invariant fashion. The time span between his work on variational calculus and the Mćanique Analytique (1788, 1808) could also be part of e the explanation—he may not have been thinking of the variational calculus when he addressed the question of a coordinate invariant formulation of the equations of motion. Section V starts by discussing the evident fact that the position and velocity at time t depend on the initial position and velocity, which can be chosen freely. We might write this as (suppressing the coordinate indices for simplicity): q = q(t, q0 , v0 ), v = v(t, q0 , v0 ), and in modern terminology we would talk about the flow in x = (q, v)-space. One problem in reading Lagrange is that he does not explicitly write the variables on which his quantities depend. In any case, he then makes an infinitesimal variation in the initial condition and looks at the corresponding variations of position and velocity at time t. In our notation: δx = (∂x/∂x0 )(t, x0 )δx0 . We would say that he considers the tangent mapping of the flow on the tangent bundle of X = T Q. Now comes the first interesting result. He makes two such variations, one denoted by δx and the other by ∆x, and he writes down a bilinear form ω(δx, ∆x), in which we recognize ω as the pull back of the canonical symplectic form on the cotangent bundle of Q, by means of the fiber derivative FL. What he then shows is that this symplectic product is constant as a function of t. This is nothing other than the invariance of the symplectic form ω under the flow in T Q. It is striking that Lagrange obtains the invariance of the symplectic form in T Q and not in T ∗ Q just as we do in the text where this is derived from Hamilton’s principle. In fact, Lagrange does not look at the equations of motion in the cotangent bundle via the transformation FL; again it is Hamilton who observes that these take the canonical Hamiltonian form. This is retrospectively puzzling since, later on in Section V, Lagrange states very explicitly that it is useful to pass to the (q, p)-coordinates by means of the coordinate transformation FL and one even sees written down a system of ordinary differential equations in Hamiltonian form, but with the total energy function H replaced by some other mysterious function −Ω. Lagrange does use the letter H for the constant value of energy, apparently in honor of Huygens. He also knew about the conservation of momentum ........................... 15 July 1998—18h02 ...........................

243. 8.2 The Geometry of Variational Principles 229 as a result of translational symmetry. The part where he does this deals with the case in which he perturbs the system by perturbing the potential from V (q) to V (q) − Ω(q), leaving the kinetic energy unchanged. To this perturbation problem, he applies his famous method of variation of constants, which is presented here in a truly nonlinear framework! In our notation, he keeps t → x(t, x0 ) as the solution of the unperturbed system, and then looks at the differential equations for x0 (t) that make t → x(t, x0 (t)) a solution of the perturbed system. The result is that, if V is the vector field of the unperturbed system and V + W is the vector field of the perturbed system, then dx0 = ((etV )∗ W )(x0 ). dt In words, x0 (t) is the solution of the time-dependent system, the vector field of which is obtained by pulling back W by means of the flow of V after time t. In the case that Lagrange considers, the dq/dt-component of the perturbation is equal to zero, and the dp/dt-component is equal ∂Ω/∂q. Thus, it is obviously in a Hamiltonian form; here one does not use anything about Legendre-transformations (which Lagrange does not seem to know). But Lagrange knows already that the flow of the unperturbed system preserves the symplectic form, and he shows that the pull back of his W under such a transformation is a vector field in Hamiltonian form. Actually, this is a time-dependent vector field, defined by the function G(t, q0 , p0 ) = −Ω(q(t, q0 , p0 )). A potential point of confusion is that Lagrange denotes this by −Ω, and writes down expressions like dΩ/dp, and one might first think these are zero because Ω was assumed to depend only on q. Lagrange presumably means that dq0 ∂G dp0 ∂G = =− . dt ∂p0 dt ∂q0 Most classical textbooks on mechanics, for example, Routh [1877, 1884], correctly point out that Lagrange has the invariance of the symplectic form in (q, v) coordinates (rather than in the canonical (q, p) coordinates). Less attention is usually paid to the variation of constants equation in Hamiltonian form, but it must have been generally known that Lagrange derived these—see, for example, Weinstein [1981]. In fact, we should point out that the whole question of linearizing the Euler–Lagrange and Hamilton equations and retaining the mechanical structure is remarkably subtle (see Marsden, Ratiu, and Raugel [1991], for example). Lagrange continues by introducing the Poisson brackets for arbitrary functions, arguing that these are useful in writing the time-derivative of arbitrary functions of arbitrary variables, along solutions of systems in Hamiltonian form. He also continues by saying that if Ω is small, then ........................... 15 July 1998—18h02 ...........................

244. 230 8. Variational Principles, Constraints, and Rotating Systems x0 (t) in zero-order approximation is a constant and he obtains the next order approximation by an integration over t; here Lagrange introduces the first steps of the so-called method of averaging. When Lagrange discovered (in 1808) the invariance of the symplectic form, the variations-of-constants equations in Hamiltonian form, and the Poisson brackets, he was already 73 years old. It is quite probable that Lagrange generously gave some of these bracket ideas to Poisson at this time. In any case, it is clear that Lagrange had a surprisingly large part of the symplectic picture of classical mechanics. Exercises 8.2-1. Derive the Hamilton–Jacobi equation starting with the phase space version of Hamilton’s principle. 8.3 Constrained Systems We begin this section with the Lagrange multiplier theorem for purposes of studying constrained dynamics. The Lagrange Multiplier Theorem. We state the theorem with a sketch of the proof, referring to Abraham, Marsden, and Ratiu [1988] for details. We shall not be absolutely precise about the technicalities (such as how to interpret dual spaces). First, consider the case of functions defined on linear spaces. Let V and Λ be Banach spaces and let ϕ : V → Λ be a smooth map. Suppose 0 is a regular value of ϕ so that C := ϕ−1 (0) is a submanifold. Let h : V → R be a smooth function and define h : V × Λ∗ → R by h(x, λ) = h(x) − λ, ϕ(x) . (8.3.1) Theorem 8.3.1 (Lagrange Multiplier Theorem for Linear Spaces). The following are equivalent conditions on x0 ∈ C: (i) x0 is a critical point of h|C; and (ii) there is a λ0 ∈ Λ∗ such that (x0 , λ0 ) is a critical point of h. Sketch of Proof. Since Dh(x0 , λ0 ) · (x, λ) = Dh(x0 ) · x − λ0 , Dϕ(x0 ) · x − λ, ϕ(x0 ) and ϕ(x0 ) = 0, the condition Dh(x0 , λ0 ) · (x, λ) = 0 is equivalent to Dh(x0 ) · x = λ0 , Dϕ(x0 ) · x (8.3.2) ........................... 15 July 1998—18h02 ...........................

245. 8.3 Constrained Systems 231 for all x ∈ V and λ ∈ Λ∗ . The tangent space to C at x0 is ker Dϕ(x0 ), so (8.3.2) implies that h|C has a critical point at x0 . Conversely, if h|C has a critical point at x0 , then Dh(x0 ) · x = 0 for all x satisfying Dϕ(x0 ) · x = 0. By the implicit function theorem, there is a smooth coordinate change that straightens out C; that is, it allows us to assume that V = W ⊕ Λ, x0 = 0, C is (in a neighborhood of 0) equal to W , and ϕ (in a neighborhood of the origin) is the projection to Λ. With these simplifications, condition (i) means that the first partial derivative of h vanishes. We choose λ0 to be D2 h(x0 ) regarded as an element of Λ∗ ; then(8.3.2) clearly holds. The Lagrange multiplier theorem is a convenient test for constrained critical points, as we know from calculus. It also leads to a convenient test for constrained maxima and minima. For instance, to test for a minimum, let α 0 be a constant, let (x0 , λ0 ) be a critical point of h, and consider hα (x, λ) = h(x) − λ, ϕ(x) + α λ − λ0 2 , (8.3.3) which also has a critical point at (x0 , λ0 ). Clearly, if hα has a minimum at (x0 , λ0 ), then h|C has a minimum at x0 . This observation is convenient since one can use the unconstrained second derivative test on hα , which leads to the theory of bordered Hessians. (For an elementary discussion, see Marsden and Tromba [1996], p.220ff.) A second remark concerns the generalization of the Lagrange multiplier theorem to the case where V is a manifold but h is still real-valued. Such a context is as follows. Let M be a manifold and let N ⊂ M be a submanifold. Suppose π : E → M is a vector bundle over M and ϕ is a section of E that is transverse to fibers. Assume N = ϕ−1 (0). Theorem 8.3.2 (Lagrange Multiplier Theorem for Manifolds). The following are equivalent for x0 ∈ N and h : M → R smooth: (i) x0 is a critical point of h|N ; and (ii) there is a section λ0 of the dual bundle E ∗ such that λ0 (x0 ) is a critical point of h : E ∗ → R defined by h(λx ) = h(x) − λx , ϕ(x) . (8.3.4) ∗ In (8.3.4), λx denotes an arbitrary element of Ex . We leave it to the reader to adapt the proof of the previous theorem to this situation. Holonomic Constraints. Many mechanical systems are obtained from higher-dimensional ones by adding constraints. Rigidity in rigid body mechanics and incompressibility in fluid mechanics are two such examples, while constraining a free particle to move on a sphere is another. Typically, constraints are of two types. Holonomic contraints are those imposed on the configuration space of a system, such as those mentioned ........................... 15 July 1998—18h02 ...........................

246. 232 8. Variational Principles, Constraints, and Rotating Systems in the preceding paragraph. Others, such as rolling constraints involve the conditions on the velocities and are termed nonholonomic. A holonomic constraint can be defined for our purposes as the specifi- cation of a submanifold N ⊂ Q of a given configuration manifold Q. (More generally a holonomic constraint is an integrable subbundle of T Q.) Since we have the natural inclusion T N ⊂ T Q, a given Lagrangian L : T Q → R can be restricted to T N to give a Lagrangian LN . We now have two La- grangian systems, namely those associated to L and to LN , assuming both are regular. We now relate the associated variational principles and the Hamiltonian vector fields. Suppose that N = ϕ−1 (0) for a section ϕ : Q → E ∗ , the dual of a vector bundle E over Q. The variational principle for LN can be phrased as δ LN (q, q) dt = 0, ˙ (8.3.5) where the variation is over curves with fixed endpoints and subject to the constraint ϕ(q(t)) = 0. By the Lagrange multiplier theorem, (8.3.5) is equivalent to δ [L(q(t), q(t)) − λ(q(t), t), ϕ(q(t)) ] dt = 0 ˙ (8.3.6) for some function λ(q, t) taking values in the bundle E and where the variation is over curves q in Q and curves λ in E.5 In coordinates, (8.3.6) reads δ [L(q i , q i ) − λa (q i , t)ϕa (q i )] dt = 0. ˙ (8.3.7) The corresponding Euler–Lagrange equations in the variables q i , λa are d ∂L ∂L ∂ϕa i = i − λa i (8.3.8) dt ∂ q ˙ ∂q ∂q and ϕa = 0. (8.3.9) They are viewed as equations in the unknowns q i (t) and λa (q i , t); if E is a trivial bundle we can take λ to be a function only of t.6 We summarize these findings as follows. 5 This conclusion assumes some regularity in t on the Lagrange multiplier λ. One can check (after the fact) that this assumption is justified by relating λ to the forces of constraint, as in the next theorem. 6 The combination L = L − λa ϕ is related to the Routhian construction for a La- a grangian with cyclic variables; see §8.9. ........................... 15 July 1998—18h02 ...........................

247. 8.3 Constrained Systems 233 Theorem 8.3.3. The Euler–Lagrange equations for LN on the manifold N ⊂ Q are equivalent to the equations (8.3.8) together with the constraints ϕ = 0. We interpret the term −λa ∂ϕa /∂q i as the force of constraint since it is the force that is added to the Euler–Lagrange operator (see §7.8) in the unconstrained space in order to maintain the constraints. In the next section we will develop the geometric interpretation of these forces of constraint. Notice that L = L − λa ϕa as a Lagrangian in q and λ is degenerate in λ; that is, the time-derivative of λ does not appear, so its conjugate momentum πa is constrained to be zero. Regarding of L as defined on T E, formally, the corresponding Hamiltonian on T ∗ E is H(q, p, λ, π) = H(q, p) + λa ϕa , (8.3.10) where H is the Hamiltonian corresponding to L. One has to be a little careful in interpreting Hamilton’s equations because L is degenerate; the general theory appropriate for this situation is the Dirac theory of constraints, which we discuss in §8.5. However, in the present context this theory is quite simple and proceeds as follows. One calls C ⊂ T ∗ E defined by πa = 0, the primary constraint set; it is the image of the Legendre transform provided the original L was regular. The canonical form Ω is pulled back to C to give a presymplectic form (a closed but possibly degenerate two-form) ΩC and one seeks XH such that iXH ΩC = dH. (8.3.11) In this case, the degeneracy of ΩC gives no equation for λ; that is, the evolution of λ is indeterminate. The other Hamiltonian equations are equivalent to (8.3.8) and (8.3.9), so in this sense the Lagrangian and Hamiltonian pictures are still equivalent. Exercises 8.3-1. Write out the second derivative of hα at (x0 , λ0 ) and relate your answer to the bordered Hessian. 8.3-2. Derive the equations for a simple pendulum using the Lagrange multiplier method and compare them with those obtained using generalized coordinates. 8.3-3 (C. Neumann [1859). (a) Derive the equations of motion of a particle of unit mass on the sphere S n−1 under the influence of a quadratic potential Aq · q, q ∈ Rn , where A is a fixed real diagonal matrix. (b) Form the matrices X = (q i q j ), P = (q i q j − q j q i ). Show that the system in ˙ ˙ ........................... 15 July 1998—18h02 ...........................

248. 234 8. Variational Principles, Constraints, and Rotating Systems ˙ ˙ (a) is equivalent to X = [P, X], P = [X, A]. (This was observed first by K. Uhlenbeck.) Equivalently, show that (−X + P λ + Aλ2 )◦ = [−X + P λ + Aλ2 , −P − Aλ]. (c) Verify that 1 1 E(X, P ) = − Tr(P 2 ) + Tr(AX) 4 2 is the total energy of this system. (d) Verify that   1  k  fk (X, P ) = Tr − Ai XAk−i + Ai P Aj P Al  , 2(k + 1)  i=0  i+j+l=k−1 i,j,l≥0 k = 1, . . . , n − 1 are conserved on the flow of the C. Neumann problem. (Ratiu[1981].) 8.4 Constrained Motion in a Potential Field We saw in the preceding section how to write the equations for a constrained system in terms of variables on the containing space. We continue this line of investigation here by specializing to the case of motion in a potential field. In fact, we shall determine by geometric methods, the extra terms that need to be added to the Euler–Lagrange equations, that is, the forces of constraint, to ensure that the constraints are maintained. Let Q be a (weak) Riemannian manifold and let N ⊂ Q be a submanifold. Let P : (T Q)|N → T N (8.4.1) be the orthogonal projection of T Q to T N defined pointwise on N . Consider a Lagrangian L : T Q → R of the form L = K − V ◦ τQ ; that is, kinetic minus potential energy. The Riemannian metric associated to the kinetic energy is denoted by , . The restriction LN = L|T N is also of the form kinetic minus potential, using the metric induced on N and the potential VN = V |N . We know from §7.7 that if EN is the energy of LN , then XEN = SN − ver( VN ), (8.4.2) ........................... 15 July 1998—18h02 ...........................

249. 8.4 Constrained Motion in a Potential Field 235 where SN is the spray of the metric on N and ver( ) denotes vertical lift. Recall that integral curves of (8.4.2) are solutions of the Euler–Lagrange equations. Let S be the geodesic spray on Q. First notice that VN and V are related in a very simple way: for q ∈ N, VN (q) = P · [ V (q)]. Thus, the main complication is in the geodesic spray. Proposition 8.4.1. SN = T P ◦ S at points of T N . Proof. For the purpose of this proof we can ignore the potential and let L = K. Let R = T Q|N , so that P : R → T N and therefore T P : T R → T (T N ), S : R → T (T Q), and T τQ ◦ S = identity since S is second-order. But T R = {w ∈ T (T Q) | T τQ (w) ∈ T N }, so S(T N ) ⊂ T R and hence T P ◦ S makes sense at points of T N . If v ∈ T Q and w ∈ Tv (T Q), then ΘL (v) · w = v, Tv τQ (w) . Letting i : R → T Q be the inclusion, we claim that P∗ ΘL|T N = i∗ ΘL . (8.4.3) Indeed, for v ∈ R and w ∈ Tv R, the deﬁnition of pull-backs gives P∗ ΘL|T N (v) · w = Pv, (T τQ ◦ T P)(w) = Pv, T (τQ ◦ P)(w) . (8.4.4) Since on R, τQ ◦ P = τQ , P∗ = P, and w ∈ Tv R, (8.4.4) becomes P∗ ΘL|T N (v) · w = Pv, T τQ (w) = v, PT τQ (w) = v, T τQ (w) = ΘL (v) · w = (i∗ ΘL )(v) · w. Taking the exterior derivative of (8.4.3) gives P∗ ΩL|T N = i∗ ΩL . (8.4.5) In particular, for v ∈ T N, w ∈ Tv R, and z ∈ Tv (T N ), the deﬁnition of pull back and (8.4.5) gives ΩL (v)(w, z) = (i∗ ΩL )(v)(w, z) = (P∗ ΩL|T N )(v)(w, z) = ΩL|T N (Pv)(T P(w), T P(z)) = ΩL|T N (v)(T P(w), z). (8.4.6) But dE(v) · z = ΩL (v)(S(v), z) = ΩL|T N (v)(SN (v), z) ........................... 15 July 1998—18h02 ...........................

250. 236 8. Variational Principles, Constraints, and Rotating Systems since S and SN are Hamiltonian vector fields for E and E|T N , respectively. From (8.4.6), ΩL|T N (v)(T P(S(v)), z) = ΩL (v)(S(v), z) = ΩL|T N (v)(SN (v), z), so by weak nondegeneracy of ΩL|T N we get the desired relation SN = T P ◦ S. Corollary 8.4.2. For v ∈ Tq N : (i) (S − SN )(v) is the vertical lift of a vector Z(v) ∈ Tq Q relative to v; (ii) Z(v) ⊥ Tq N ; and (iii) Z(v) = − v v + P( v v) is minus the normal component of v v, where in v v, v is extended to a vector field on Q tangent to N . Proof. (i) Since T τQ (S(v)) = v = T τQ (SN (v)), we have T τQ (S − SN )(v) = 0, that is, (S − SN )(v) is vertical. The statement now follows from the comments following Definition 7.7.1. (ii) For u ∈ Tq Q, we have T P · ver(u, v) = ver(Pu, v) since d d ver(Pu, v) = (v + tPu) = P(v + tu) dt t=0 dt t=0 = T P · ver(u, v). (8.4.7) By Part (i), S(v) − SN (v) = ver(Z(v), v) for some Z(v) ∈ Tq Q, so that using the previous theorem, (8.4.7), and P ◦ P = P, we get ver(PZ(v), v) = T P · ver(Z(v), v) = T P(S(v) − SN (v)) = T P(S(v) − T P ◦ S(v)) = 0. Therefore, PZ(v) = 0, that is, Z(v) ⊥ Tq N . (iii) Let v(t) be a curve of tangents to N ; v(t) = c(t), where c(t) ∈ N . Then ˙ in a chart, S(c(t), v(t)) = c(t), v(t), v(t), γc(t) (v(t), v(t)) ........................... 15 July 1998—18h02 ...........................

251. 8.4 Constrained Motion in a Potential Field 237 by (7.5.5). Extending v(t) to a vector field v on Q tangent to N we get, in a standard chart, dv vv = −γc (v, v) + Dv(c) · v = −γc (v, v) + dt by (7.5.19), so on T N , dv S(v) = − ver( v v, v). dt Since dv/dt ∈ T N , (8.4.7) and the previous proposition give dv dv SN (v) = T P − ver(P( v v), v) = − ver(P( v v), v). dt dt Thus, by part (i), ver(Z(v), v) = S(v) − SN (v) = ver(− vv +P v v, v). The map Z : T N → T Q is called the force of constraint. We shall prove below that if the codimension of N in Q is one, then Z(v) = − vv + P( v v) =− v v, n n, where n is the unit normal vector field to N in Q, equals the negative of the quadratic form associated to the second funamental form of N in Q, a result due to Gauss. (We shall define the second fundamental form, which measures how “curved” N is within Q, shortly.) It is not obvious at first that the expression P( v v) − v v depends only on the pointwise values of v, but this follows from its identification with Z(v). To prove the above statement, we recall that the Levi–Civita covariant derivative has the property that for vector fields u, v, w ∈ X(Q) the following identity is satisfied: w[ u, v ] = w u, v + u, wv , (8.4.8) as may be easily checked. Assume now that u an v are vector fields tangent to N and n is the unit normal vector field to N in Q. The identity (8.4.8) yields v u, n + u, vn = 0. (8.4.9) The second fundamental form in Riemannian geometry is defined to be the map (u, v) → − u n, v (8.4.10) ........................... 15 July 1998—18h02 ...........................

252. 238 8. Variational Principles, Constraints, and Rotating Systems with u, v, n as above. It is a classical result that this bilnear form is symmetric and hence is uniquely determined by polarization from its quadratic form − v n, v . In view of equation (8.4.9), this quadratic from has the al- ternate expression v v, n which, after multiplcation by n, equals −Z(v), thereby proving the claim above. As indicated, this discussion of the second fundamental form is under the assumption that the codimension of N in Q is one—keep in mind that our discussion of forces of constraint requires no such restriction. As before, interpret Z(v) as the constraining force needed to keep particles in N . Notice that N is totally geodesic (that is, geodesics in N are geodesics in Q) iﬀ Z = 0. Mention Rubin Ungar Exercises 8.4-1. Compute the force of constraint Z and the second fundamental form for the sphere of radius R in R3 . 8.4-2. Assume L is a regular Lagrangian on T Q and N ⊂ Q. Let i : T N → T Q be the embedding obtained from N ⊂ Q and let ΩL be the Lagrange two-form on T Q. Show that i∗ ΩL is the Lagrange two-form ΩL|T N on T N . Assuming L is hyperregular, show that the Legendre transform deﬁnes a symplectic embedding T ∗ N ⊂ T ∗ Q. 8.4-3. In R3 , let 1 H(q, p) = p 2 − (p · q)2 + mgq 3 , 2m where q = (q 1 , q 2 , q 3 ). Show that Hamilton’s equations in R3 automatically preserve T ∗ S 2 and give the pendulum equations when restricted to this invariant (symplectic) submanifold. (Hint: Use the formulation of Lagrange’s equations with constraints in §§8.3) 8.4-4. Redo the C. Neumann problem in Exercise 8.3-3 using Corol- lary 8.4.2 and the interpretation of the constraining force in terms of the second fundamental form. 8.5 Dirac Constraints If (P, Ω) is a symplectic manifold, a submanifold S ⊂ P is called a symplectic submanifold when ω := i∗ Ω is a symplectic form on S, i : S → P being the inclusion. Thus, S inherits a Poisson bracket structure; its relationship to the bracket structure on P is given by a formula of Dirac [1950] that will be derived in this section. Dirac’s work was motivated by the study of constrained systems, especially relativistic ones, where one thinks of S as a constraint subspace of phase space (see Gotay, Isenberg, and ........................... 15 July 1998—18h02 ...........................

253. 8.5 Dirac Constraints 239 Marsden [1998] and references therein for more information). Let us work in the finite-dimensional case; the reader is invited to study the intrinsic infinite-dimensional version using Remark 1 below. Dirac’s formula. Let dim P = 2n and dim S = 2k. In a neighborhood of a point z0 of S, choose coordinates z 1 , . . . , z 2n on P such that S is given by z 2k+1 = 0, . . . , z 2n = 0 and so z 1 , . . . , z 2k provide local coordinates for S. Consider the matrix whose entries are C ij (z) = {z i , z j }, i, j = 2k + 1, . . . , 2n. Assume that the coordinates are chosen so that C ij is an invertible matrix at z0 and hence in a neighborhood of z0 . (Such coordinates always exist, as is easy to see.) Let its inverse be denoted [Cij (z)]. Let F be a smooth function on P and F |S its restriction to S. We are interested in relating XF |S and XF as well as the brackets {F, G}|S and {F |S, G|S}. Proposition 8.5.1 (Dirac’s Bracket Formula). In a coordinate neighborhood as described above, and for z ∈ S, we have 2n XF |S (z) = XF (z) − {F, z i }Cij (z)Xzj (z) (8.5.1) i,j=2k+1 and 2n {F |S, G|S}(z) = {F, G}(z) − {F, z i }Cij (z){z j , G}. (8.5.2) i,j=2k+1 Proof. To verify (8.5.1), we show that the right-hand side satisfies the condition required for XF |S (z), namely that it be a vector field on S and that ωz (XF |S (z), v) = d(F |S)z · v (8.5.3) for v ∈ Tz S. Since S is symplectic, Tz S ∩ (Tz S)Ω = {0}, where (Tz S)Ω denotes the Ω-orthogonal complement. Since dim(Tz S) + dim(Tz S)Ω = 2n, we get Tz P = Tz S ⊕ (Tz S)Ω . (8.5.4) ........................... 15 July 1998—18h02 ...........................

254. 240 8. Variational Principles, Constraints, and Rotating Systems If πz : Tz P → Tz S is the associated projection operator one can verify that XF |S (z) = πz · XF (z), (8.5.5) so, in fact, (8.5.1) is a formula for πz in coordinates; equivalently, 2n (Id −πz )XF (z) = {F, z i }Cij (z)Xzj (z) (8.5.6) i,j=2k+1 gives the projection to (Tz S)Ω . To verify (8.5.6), we need to check that the right-hand side (i) is an element of (Tz S)Ω ; (ii) equals XF (z) if XF (z) ∈ (Tz S)Ω ; and (iii) equals 0 if XF (z) ∈ Tz S. To prove (i), observe that XK (z) ∈ (Tz S)Ω means Ω(XK (z), v) = 0 for all v ∈ Tz S; that is, dK(z) · v = 0 for all v ∈ Tz S. But for K = z , j = 2k + 1, . . . , 2n, K ≡ 0 on S, and hence dK(z) · v = 0. j Thus, Xzj (z) ∈ (Tz S)Ω , so (i) holds. For (ii), if XF (z) ∈ (Tz S)Ω , then dF (z) · v = 0 for all v ∈ Tz S and, in particular, for v = ∂/∂z i , i = 1, . . . , 2k. Therefore, for z ∈ S, we can write 2n dF (z) = aj dz j (8.5.7) j=2k+1 and hence 2n XF (z) = aj Xzj (z). (8.5.8) j=2k+1 The aj are determined by pairing (8.5.8) with dz i , i = 2k + 1, . . . , 2n, to give 2n 2n − dz i , XF (z) = {F, z i } = aj {z j , z i } = aj C ji , j=2k+1 j=2k+1 ........................... 15 July 1998—18h02 ...........................

255. 8.5 Dirac Constraints 241 or 2n aj = {F, z i }Cij , (8.5.9) i=2k+1 which proves (ii). Finally, for (iii), XF (z) ∈ Tz S = ((Tz S)Ω )Ω means XF (z) is Ω orthogonal to each Xzj , j = 2k + 1, . . . , 2n. Thus, {F, z j } = 0, so the right-hand side of (8.5.6) vanishes. Formula (8.5.6) is therefore proved, and so, equivalently (8.5.1) holds. Formula (8.5.2) follows by writing {F |S, G|S} = ω(XF |S , XG|S ) and substituting (8.5.1). In doing this, the last two terms cancel. In (8.5.2) notice that {F |S, G|S}(z) is intrinsic to F |S, G|S, and S. The bracket does not depend on how F |S and G|S are extended oﬀ S to functions F, G on P . This is not true for just {F, G}(z), which does depend on the extensions, but the extra term in (8.5.2) cancels this dependence. Remarks. 1. A coordinate-free way to write (8.5.2) is as follows. Write S = ψ −1 (m0 ), where ψ : P → M is a submersion on S. For z ∈ S, and m = ψ(z), let ∗ ∗ C m : T m M × Tm M → R (8.5.10) be given by Cm (dFm , dGm ) = {F ◦ ψ, G ◦ ψ}(z) (8.5.11) for F, G ∈ F(M ). Assume Cm is invertible, with “inverse” −1 Cm : Tm M × Tm M → R. Then −1 {F |S, G|S}(z) = {F, G}(z) − Cm (Tz ψ · XF (z), Tz ψ · XG (z)). (8.5.12) 2. There is another way to derive and write Dirac’s formula using complex structures. Suppose , z is an inner product on Tz P and J z : Tz P → Tz P is an orthogonal transformation satisfying J2 = − Identity and, as in §5.3, z Ωz (u, v) = Jz u, v (8.5.13) ........................... 15 July 1998—18h02 ...........................

256. 242 8. Variational Principles, Constraints, and Rotating Systems for all u, v ∈ Tz P . With the inclusion i : S → P as before, we get corresponding structures induced on S; let ω = i∗ Ω. (8.5.14) If ω is nondegenerate, then (8.5.14) and the induced metric deﬁnes an associated complex structure K on S. At a point z ∈ S, suppose one has arranged to choose Jz to map Tz S to itself, and that Kz is the restriction of Jz to Tz S. At z, we then get (Tz S)⊥ = (Tz S)Ω and thus symplectic projection coincides with orthogonal projection. From (8.5.5), and using coordinates as described earlier, but for which the Xzj (z) are also orthogonal, we get 2n XF |S (z) = XF (z) − XF (z), Xzj (z) Xzj (z) j=2k+1 2n = XF (z) + Ω(XF (z), J−1 Xzj (z))Xzj . (8.5.15) j=2k+1 This is equivalent to (8.5.1) and so also gives (8.5.2); to see this, one shows that 2n J−1 Xzj (z) = − Xzi (z)Cij (z). (8.5.16) i=2k+1 p Indeed, the symplectic pairing of each side with Xzp gives δj . 3. For a relationship between Poisson reduction and Dirac’s formula, see Marsden and Ratiu [1986]. Examples (a) Holonomic Constraints. To treat holonomic constraints by the Dirac formula, proceed as follows. Let N ⊂ Q be as in §8.4, so that T N ⊂ T Q; with i : N → Q the inclusion, one ﬁnds (T i)∗ ΘL = ΘLN by considering the following commutative diagram: Ti T N − − − − → T Q|N −−−−   FL   N FL T ∗ N ← − − − − T ∗ Q|N −−−− projection This realizes T N as a symplectic submanifold of T Q and so Dirac’s formula can be applied, reproducing (8.4.2). See Exercise 8.4-2. ........................... 15 July 1998—18h02 ...........................

257. 8.5 Dirac Constraints 243 (b) KdV Equation. Suppose7 one starts with a Lagrangian of the form L(vq ) = α(q), v − h(q), (8.5.17) where α is a one-form on Q and h is a function on Q. In coordinates, (8.5.17) reads L(q i , q i ) = αi (q)q i − h(q i ). ˙ ˙ (8.5.18) The corresponding momenta are ∂L pi = = αi ; i.e., p = α(q), (8.5.19) ∂ qi ˙ while the Euler–Lagrange equations are d ∂L ∂αj j ∂h (αi (q j )) = i = q − i, ˙ dt ∂q ∂q i ∂q that is, ∂αi j ∂αj j ∂h q − ˙ q = − i. ˙ (8.5.20) ∂q j ∂q i ∂q In other words, with v i = q i , ˙ iv dα = −dh. (8.5.21) If dα is nondegenerate on Q then (8.5.21) defines Hamilton’s equations for a vector field v on Q with Hamiltonian h and symplectic form Ωα = −dα. This collapse, or reduction, from T Q to Q is another instance of the Dirac theory and how it deals with degenerate Lagrangians in attempting to form the corresponding Hamiltonian system. Here the primary constraint manifold is the graph of α. Note that if we form the Hamiltonian on the primaries, H = pi q i − L = αi q i − αi q i + h(q) = h(q), ˙ ˙ ˙ (8.5.22) that is, H = h, as expected from (8.5.21). To put the KdV equation ut +6uux +uxxx = 0 in this context, let u = ψx ; that is, ψ is an indefinite integral for u. Observe that the KdV equation is the Euler–Lagrange equation for L(ψ, ψt ) = 1 2 ψt ψ x + ψx − 1 (ψxx )2 dx, 3 2 (8.5.23) 7 We thank P. Morrison and M. Gotay for the following comment on how to view the KdV equation using constraints; see Gotay [1988]. ........................... 15 July 1998—18h02 ...........................

258. 244 8. Variational Principles, Constraints, and Rotating Systems that is, δ L dt = 0 gives ψxt + 6ψx ψxx + ψxxxx = 0 which is the KdV equation for u. Here α is given by 1 α(ψ), ϕ = 2 ψx ϕ dx (8.5.24) and so by formula 6 in the table in §4.4, −dα(ψ)(ψ1 , ψ2 ) = 1 2 (ψ1 ψ2x − ψ2 ψ1x ) dx (8.5.25) which equals the KdV symplectic structure (3.2.9). Moreover, (8.5.22) gives the Hamiltonian H= 1 2 (ψxx ) 2 − ψx dx = 3 1 2 (ux ) 2 − u3 dx (8.5.26) also coinciding with Example (c) of §3.2. Exercises 8.5-1. Derive formula (8.4.2) from (8.5.1). 8.5-2. Work out Dirac’s formula for (a) T ∗ S 1 ⊂ T ∗ R2 and (b) T ∗ S 2 ⊂ T ∗ R3 In each case, note that the embedding makes use of the metric. Reconcile your analysis with what you found in Exercise 8.4-2. 8.6 Centrifugal and Coriolis Forces In this section we discuss, in an elementary way, the basic ideas of centrifugal and Coriolis forces. This section takes the view of rotating observers while the next sections take the view of rotating systems. Rotating Frames. Let V be a three-dimensional oriented inner product space that we regard as “inertial space.” Let ψt be a curve in SO(V ), the group of orientation-preserving orthogonal linear transformations of V to V , and let Xt be the (possibly time-dependent) vector ﬁeld generating ψt ; that is, d Xt (ψt (v)) = ψt (v), (8.6.1) dt or, equivalently, ˙ −1 Xt (v) = (ψt ◦ ψt )(v). (8.6.2) ........................... 15 July 1998—18h02 ...........................

259. 8.6 Centrifugal and Coriolis Forces 245 Differentation of the orthogonality conditon ψt · ψt = Id shows that Xt is T skew symmetric. A vector ω in three space defines a skew symmetric 3 × 3 linear transfor- ˆ mation ω using the cross product; specifically, it is defined by the equation ω(v) = ω × v. ˆ Conversely, any skew matrix can be so represented in a unique way. As we shall see later (see §9.2, especially equation (9.2.4)) that this is a fundamental link between the Lie algebra of the rotation group and the cross product. This relation also will play a crucial role in the dynamics of a rigid body. In particular, we can represent the skew matrix Xt this way: Xt (v) = ω(t) × v, (8.6.3) which defines ω(t), the instantaneous rotation vector . Let {e1 , e2 , e3 } be a fixed (inertial) orthonormal frame in V and let {ξ i = ψt (ei ) | i = 1, 2, 3} be the corresponding rotating frame. Given a point v ∈ V , let q = (q 1 , q 2 , q 3 ) denote the vector in R3 defined by v = q i ei and let qR ∈ R3 be the corresponding coordinate vector representing the i components of the same vector v in the rotating frame, so v = qR ξ i . Let At = A(t) be the matrix of ψt relative to the basis ei , that is, ξ i = Aj ej ;i then q = At qR ; i.e., q j = Aj qR , i i (8.6.4) and (8.6.2) in matrix notation becomes ω = At A−1 . ˆ ˙ t (8.6.5) Newton’s Law in a Rotating Frame. Assume that the point v(t) moves in V according to Newton’s second law with a potential energy U (v). Using U (q) for the corresponding function induced on R3 , Newton’s law reads m¨ = − U (q), q (8.6.6) which are the Euler–Lagrange equations for m ˙ L(q, q) = q, q − U (q) ˙ ˙ (8.6.7) 2 or Hamilton’s equations for 1 H(q, p) = p, p + U (q). (8.6.8) 2m ........................... 15 July 1998—18h02 ...........................

260. 246 8. Variational Principles, Constraints, and Rotating Systems To find the equation satisfied by qR , differentiate (8.6.4) with respect to time q = At qR + At qR = At A−1 q + At qR , ˙ ˙ ˙ ˙ t ˙ (8.6.9) that is, q = ω(t) × q + At qR , ˙ ˙ (8.6.10) where, by abuse of notation, ω is also used for the representation of ω in the inertial frame ei . Differentiating (8.6.10), ˙ ˙ ˙ ˙ q = ω × q + ω × q + At qR + At qR ¨ ¨ = ω × q + ω × (ω × q + At qR ) + At A−1 At qR + At qR , ˙ ˙ ˙ t ˙ ¨ that is, q = ω × q + ω × (ω × q) + 2(ω × At qR ) + At qR . ¨ ˙ ˙ ¨ (8.6.11) The angular velocity in the rotating frame is (see (8.6.4)): ω R = A−1 ω, t i.e., ω = At ω R . (8.6.12) Differentiating (8.6.12) with respect to time gives ω = At ω R + At ω R = At A−1 ω + At ω R = At ω R , ˙ ˙ ˙ ˙ t ˙ ˙ (8.6.13) since At A−1 ω = ω × ω = 0. Multiplying (8.6.11) by A−1 gives ˙ t t A−1 q = ω R × qR + ω R × (ω R × qR ) + 2(ω R × qR ) + qR . t ¨ ˙ ˙ ¨ (8.6.14) Since m¨ = − U (q), we have q mA−1 q = − UR (qR ), t ¨ (8.6.15) where the rotated potential UR is the time-dependent potential defined by UR (qR , t) = U (At qR ) = U (q), (8.6.16) so that U (q) = At UR (qR ). Therefore, by (8.6.15), Newton’s equations (8.6.6) become m¨ R + 2(ω R × mqR ) + mω R × (ω R × qR ) + mω R × qR q ˙ ˙ = − UR (qR , t), that is, m¨ R = − UR (qR , t) − mω R × (ω R × qR ) q − 2m(ω R × qR ) − mω R × qR , ˙ ˙ (8.6.17) which expresses the equations of motion entirely in terms of rotated quantities. ........................... 15 July 1998—18h02 ...........................

261. 8.6 Centrifugal and Coriolis Forces 247 Ficticious Forces. There are three types of “fictitious forces” that suggest themselves if we try to identify (8.6.17) with ma = F: (i) centrifugal force mω R × (qR × ω R ); (ii) Coriolis force 2mqR × ω R ; and ˙ (iii) Euler force mqR × ω R . ˙ Note that the Coriolis force 2mω R × qR is orthogonal to ω R and mqR ˙ ˙ while the centrifugal force mω R × (ω R × qR ) = m[(ω R · qR )ω R − ω R 2 qR ] is in the plane of ω R and qR . Also note that the Euler force is due to the nonuniformity of the rotation rate. Lagrangian Form. It is of interest to ask the sense in which (8.6.17) is Lagrangian or Hamiltonian. To answer this, it is useful to begin with the Lagrangian approach, which, we will see, is simpler. Substitute (8.6.10) into (8.6.7) to express the Lagrangian in terms of rotated quantities: m L= ω × q + At qR , ω × q + At qR − U (q) ˙ ˙ 2 m = ω R × qR + qR , ω R × qR + qR − UR (qR , t), ˙ ˙ (8.6.18) 2 ˙ which defines a new (time-dependent!) Lagrangian LR (qR , qR , t). Remark- ably, (8.6.17) are precisely the Euler–Lagrange equations for LR ; that is, (8.6.17) are equivalent to d ∂LR ∂LR i = , ˙ dt ∂ qR ∂qi R as is readily verified. If one thinks about performing a time-dependent transformation in the variational principle, then in fact, one sees that this is reasonable. Hamiltonian Form. To find the sense in which (8.6.17) is Hamiltonian, perform a Legendre transformation on LR . The conjugate momentum is ∂LR pR = = m(ω R × qR + qR ) ˙ (8.6.19) ˙ ∂ qR and so the Hamiltonian has the expression HR (qR , pR ) = pR , qR − LR ˙ 1 1 = pR , pR − mω R × qR − pR , pR + UR (qR , t) m 2m 1 = pR , pR + UR (qR , t) − pR , ω R × qR . (8.6.20) 2m ........................... 15 July 1998—18h02 ...........................

262. 248 8. Variational Principles, Constraints, and Rotating Systems Thus, (8.6.17) are equivalent to Hamilton’s canonical equations with Hamil- tonian (8.6.20) and with the canonical symplectic form. In general, HR is time-dependent. Alternatively, if we perform the momentum shift pR = pR − mω R × qR = mqR , ˙ (8.6.21) then we get ˜ HR (qR , pR ) : = HR (qR , pR ) 1 m = pR , pR + UR (qR ) − ω R × qR 2 , (8.6.22) 2m 2 which is in the usual form of kinetic plus potential energy, but now the potential is amended by the centrifugal potential m ω R × qR 2 /2 and the canonical symplectic structure Ωcan = dqi ∧ d(pR )i R gets transformed, by the momentum shifting lemma, or directly, to dqi ∧ dpRi = dqi ∧ dpRi + R R i i ijk ωR dqR ∧ dqj , R where ijk is the alternating tensor. Note that ˜ ˜ ΩR = Ωcan + ∗ω R , (8.6.23) where ∗ω R means the two-form associated to the vector ω R and that (8.6.23) has the same form as the corresponding expression for a particle in a magnetic field (§6.7). In general, the momentum shift (8.6.21) is time-dependent, so care is needed in interpreting the sense in which the equations for pR and qR are Hamiltonian. In fact, the equations should be computed as follows. Let XH be a Hamiltonian vector field on P and let ζt : P → P be a time-dependent map with generator Yt : d ζt (z) = Yt (ζt (z)). (8.6.24) dt Assume that ζt is symplectic for each t. If z(t) = XH (z(t)) and we let ˙ w(t) = ζt (z(t)), then w satisfies w = T ζt · XH (z(t)) + Yt (ζt (z(t)), ˙ (8.6.25) that is, w = XK (w) + Yt (w) ˙ (8.6.26) −1 where K = H ◦ ζt . The extra term Yt in (8.6.26) is, in the example under consideration, the Euler force. ........................... 15 July 1998—18h02 ...........................

263. 8.7 The Geometric Phase for a Particle in a Hoop 249 So far we have been considering a fixed system as seen from different rotating observers. Analogously, one can consider systems that themselves are subjected to a superimposed rotation, an example being the Foucault pendulum. It is clear that the physical behavior in the two cases can be different—in fact, the Foucault pendulum and the example in the next section show that one can get a real physical effect from rotating a system— obviously rotating observers can cause nontrivial changes in the description of a system, but cannot make any physical difference. Nevertheless, the strategy for the analysis of rotating systems is analogous to the above. The easiest approach, as we have seen, is to transform the Lagrangian. The reader may wish to reread §2.10 for an easy and specific instance of this. Exercises 8.6-1. Generalize the discussion of Newton’s law seen in a rotating frame to that of a particle moving in a magnetic field as seen from a rotating observer. Do so first directly and then by Lagrangian methods. 8.7 The Geometric Phase for a Particle in a Hoop This discussion follows Berry [1985] with some small modifications (due to Marsden, Montgomery, and Ratiu [1990]) necessary for a geometric interpretation of the results. Figure 8.7.1, shows a planar hoop (not necessarily circular) in which a bead slides without friction. k Rθ q'(s) α q(s) Rθ q(s) s Rθ q'(s) Figure 8.7.1. A particle sliding in a rotating hoop. As the bead is sliding, the hoop is rotated in its plane through an angle ˙ θ(t) with angular velocity ω(t) = θ(t)k. Let s denote the arc length along the hoop, measured from a reference point on the hoop and let q(s) be the vector from the origin to the corresponding point on the hoop; thus the shape of the hoop is determined by this function q(s). The unit tangent ........................... 15 July 1998—18h02 ...........................

264. 250 8. Variational Principles, Constraints, and Rotating Systems vector is q (s) and the position of the reference point q(s(t)) relative to an inertial frame in space is Rθ(t) q(s(t)), where Rθ is the rotation in the plane of the hoop through an angle θ. Note that ˙ −1 Rθ Rθ q = ω × q and Rθ ω = ω. The Equations of Motion. The conﬁguration space is a ﬁxed closed curve (the hoop) in the plane with length . The Lagrangian L(s, s, t) is ˙ simply the kinetic energy of the particle. Since d Rθ(t) q(s(t)) = Rθ(t) q (s(t))s(t) + Rθ(t) [ω(t) × q(s(t))], ˙ dt the Lagrangian is 1 L(s, s, t) = ˙ m q (s)s + ω × q 2 . ˙ (8.7.1) 2 Note that the momentum conjugate to s is p = ∂L/∂ s; that is, ˙ p = mq · [q s + ω × q] = mv, ˙ (8.7.2) where v is the component of the velocity with respect to the inertial frame tangent to the curve. The Euler–Lagrange equations d ∂L ∂L = dt ∂ s ˙ ∂s become d [q · (q s + ω × q)] = (q s + ω × q) · (q s + ω × q ). ˙ ˙ ˙ dt Using q 2 = 1, its consquence q · q = 0, and simplifying, we get s + q · (ω × q) − (ω × q) · (ω × q ) = 0. ¨ ˙ (8.7.3) The second and third terms in (8.7.3) are the Euler and centrifugal forces, ˙ respectively. Since ω = θk, we can rewrite (8.7.3) as ¨ ˙ ¨ s = θ2 q · q − θq sin α, (8.7.4) where α is as in Figure 8.7.1 and q = q . Averaging. From (8.7.4) and Taylor’s formula with remainder, we get t s(t) = s0 + s0 t + ˙ ˙ (t − τ ){θ(τ )2 q(s(τ )) · q (s(τ )) 0 ¨ − θ(τ )q(s(τ )) sin α(s(τ ))} dτ. (8.7.5) ........................... 15 July 1998—18h02 ...........................

265. 8.7 The Geometric Phase for a Particle in a Hoop 251 ˙ ¨ The angular velocity θ and acceleration θ are assumed small with respect to the particle’s velocity, so by the averaging theorem (see, for example, Hale [1963]), the s-dependent quantities in (8.7.5) can be replaced by their averages round the hoop: t ˙ 1 s(t) ≈ s0 + s0 t + ˙ (t − τ ) θ(τ )2 q · q ds 0 0 ¨ 1 −θ(τ ) q(s) sin α(s) ds dτ. (8.7.6) 0 Technical Aside. The essence of averaging in this case can be seen as follows. Suppose g(t) is a rapidly varying function whose oscillations are bounded in magnitude by a constant C and f (t) is slowly varying on an interval [a, b]. Over one period of g, say [α, β], we have β β f (t)g(t) dt ≈ g f (t) dt, (8.7.7) α α where β 1 g= g(t) dt β−α α is the average of g. The assumption that the oscillations of g are bounded by C means that |g(t) − g| ≤ C for all t ∈ [α, β]. β The error in (8.7.7) is α f (t)(g(t) − g) dt, whose absolute value is bounded as follows. Let M be the maximum value of f on [α, β] and m be the minimum. Then β β f (t)[g(t) − g]dt = (f (t) − m)[g(t) − g]dt α α ≤ (β − α)(M − m)C ≤ (β − α)2 DC, where D is the maximum of |f (t)| for α ≤ t ≤ β. Now these errors over each period are added up over [a, b]. Since the error estimate has the square of β − α as a factor, one still gets something small as the period of g tends to 0. In (8.7.5) we change variables from t to s, do the averaging, and then change back. ........................... 15 July 1998—18h02 ...........................

266. 252 8. Variational Principles, Constraints, and Rotating Systems The Phase Formula. The first inner integral in (8.7.6) over s vanishes d (since the integrand is ds q(s) 2 ) and the second is 2A where A is the area enclosed by the hoop. Integrating by parts, T T ¨ ˙ (T − τ )θ(τ ) dτ = −T θ(0) + ˙ ˙ θ(τ ) dτ = −T θ(0) + 2π (8.7.8) 0 0 assuming the hoop makes one complete revolution in time T . Substituting (8.7.8) in (8.7.6) gives 2A ˙ 4πA s(T ) ≈ s0 + s0 T + ˙ θ0 T − , (8.7.9) ˙ ˙ where θ0 = θ(0). The initial velocity of the bead relative to the hoop is s0 , ˙ while its component along the curve relative to the inertial frame is (see (8.7.2)), v0 = q (0) · [q (0)s0 + ω 0 × q(0)] = s0 + ω 0 q(s0 ) sin α(s0 ). ˙ ˙ (8.7.10) Now we replace s0 in (8.7.9) by its expression in terms of v0 from (8.7.10) ˙ and average over all initial conditions to get 4πA s(T ) − s0 − v0 T = − , (8.7.11) which means that on average, the shift in position is by 4πA/ between the ˙ rotated and nonrotated hoop. Note that if θ0 = 0 (the situation assumed by Berry [1985]), then averaging over initial conditions is not necessary. This extra length 4πA/ is sometimes called the geometric phase or the Berry-Hannay phase. This example is related to a number of interesting effects, both classically and quantum mechanically, such as the Foucault pendulum and the Aharonov-Bohm effect. The effect is known as holonomy and can be viewed as an instance of reconstruction in the context of symmetry and reduction. For further information and additional references, see Aharonov and Anandan[1987], Montgomery [1988], [1990], and Marsden, Montgomery, and Ratiu [1989, 1990]. For related ideas in soliton dynamics, see Alber and Marsden [1992]. Exercises 8.7-1. Consider the dynamics of a ball in a slowly rotating planar hoop, as in the text. However, this time, consider rotating the hoop about an axis that is not perpendicular to the plane of the hoop, but makes an angle θ with the normal. Compute the geometric phase for this problem. 8.7-2. Study the geometric phase for a particle in a general spatial hoop that is moved through a closed curve in SO(3). ........................... 15 July 1998—18h02 ...........................

267. 8.8 Moving Systems 253 8.7-3. Consider the dynamics of a ball in a slowly rotating planar hoop, as in the text. However, this time, consider a charged particle with charge e and a fixed magnetic field B = × A in the vicinity of the hoop. Compute the geometric phase for this problem. 8.8 Moving Systems The particle in the rotating hoop is an example of a rotated or, more generally, a moving system. Other examples are a pendulum on a merry- go-round (Exercise 8.8-4) and a fluid on a rotating sphere (like the Earth’s ocean and atmosphere). As we have emphasized, systems of this type are not to be confused with rotating observers! Actually rotating a system causes real physical effects, such as the trade winds and hurricanes. This section develops a general context for such systems. Our purpose is to show how to systematically derive Lagrangians and the resulting equations of motion for moving systems, like the bead in the hoop of the last section. This will also set up the reader who wants to pursue the question of how moving systems fit in the context of phases (Marsden, Montgomery, and Ratiu [1990]). The Lagrangian. Consider a Riemannian manifold S, a submanifold Q, and a space M of embeddings of Q into S. Let mt ∈ M be a given curve. If a particle in Q is following a curve q(t), and if Q moves by superposing the motion mt , then the path of the particle in S is given by mt (q(t)). Thus, its velocity in S is given by Tq(t) mt · q(t) + Zt (mt (q(t))), ˙ (8.8.1) where Zt (mt (q)) = dt mt (q). Consider a Lagrangian on T Q of the usual d form of kinetic minus potential energy: Lmt (q, v) = 1 2 Tq(t) mt · v + Zt (mt (q)) 2 − V (q) − U (mt (q)), (8.8.2) where V is a given potential on Q, and U is a given potential on S. The Hamiltonian. We now compute the Hamiltonian associated to this Lagrangian by taking the associated Legendre transform. If we take the derivative of (8.8.2) with respect to v in the direction of w, we obtain: ∂Lmt T · w = p · w = Tq(t) mt · v + Zt (mt (q(t))) , Tq(t) mt · w ∂v mt (q(t)) (8.8.3) ∗ where p · w means the natural pairing between the covector p ∈ Tq(t) Q and the vector w ∈ Tq(t) Q, , mt (q(t)) denotes the metric inner product on S at the point mt (q(t)) and T denotes the orthogonal projection to the ........................... 15 July 1998—18h02 ...........................

268. 254 8. Variational Principles, Constraints, and Rotating Systems tangent space T mt (Q) using the metric of S at mt (q(t)). We endow Q with the (possible time dependent) metric induced by the mapping mt . In other words, we choose the metric on Q that makes mt into an isometry for each t. Using this definition, (8.8.3) gives: −1 T p · w = v + Tq(t) mt · Zt (mt (q(t))) , w ; q(t) that is, −1 T p = v + Tq(t) mt · Zt (mt (q(t)) , (8.8.4) where is the index lowering operation at q(t) using the metric on Q. Physically, if S is R3 , then p is the inertial momentum (see the hoop example in the preceding section). This extra term Zt (mt (q))T is associated with a connection called the Cartan connection on the bundle Q × M → M , with horizontal lift defined to be Z(m) → (T m−1 · Z(m)T , Z(m)). (See for example, Marsden and Hughes [1983] for an account of some aspects of Cartan’s contributions.) The corresponding Hamiltonian (given by the standard prescription H = pv − L) picks up a cross-term and takes the form ⊥ Hmt (q, p) = 1 2 p 2 − P(Zt ) − 1 2 Zt 2 + V (q) + U (mt (q)), (8.8.5) where the time dependent vector field Zt on Q is defined by −1 Zt (q) = Tq(t) mt · [Zt (mt (q)]T ⊥ and where P(Zt (q))(q, p) = p, Zt (q) and Zt denotes the component perpendicular to mt (Q). The Hamiltonian vector field of this cross-term, namely XP(Zt ) , represents the non-inertial forces and also has the natural interpretation as a horizontal lift of the vector field Zt relative to a certain connection on the bundle T ∗ Q × M → M , naturally derived from the Cartan connection. Remarks on Averaging. Let G be a Lie group which acts on T ∗ Q in a Hamiltonian fashion and leaves H0 (defined by setting Z = 0 and U = 0 in (8.8.5)) invariant. (Lie groups are discussed in the next chapter, so these remarks can be omitted on a first reading.) In our examples, G is either R acting on T ∗ Q by the flow of H0 (the hoop), or a subgroup of the isometry group of Q which leaves V and U invariant, and acts on T ∗ Q by cotangent lift (this is appropriate for the Foucault pendulum). In any case, we assume G has an invariant measure relative to which we can average. Assuming the “averaging principle” (see Arnold [1989], for example) we replace Hmt by its G-average, ⊥ Hmt (q, p) = 1 2 p 2 − P(Zt ) − 1 2 Zt 2 + V (q) + U (mt (q)) . (8.8.6) ........................... 15 July 1998—18h02 ...........................

269. 8.8 Moving Systems 255 ⊥ In (8.8.6) we shall assume the term 1 2 Zt 2 is small and discard it. Thus, define H(q, p, t) = 1 p 2 − P(Zt ) + V (q) + U (mt (q)) 2 = H0 (q, p) − P(Zt ) + U (mt (q)) . (8.8.7) Consider the dynamics on T ∗ Q × M given by the vector field (XH , Zt ) = (XH0 − X P(Zt ) +X U ◦mt , Zt ). (8.8.8) The vector field, consisting of the extra terms in this representation due to the superposed motion of the system, namely hor(Zt ) = (−X P(Zt ) , Zt ), (8.8.9) has a natural interpretation as the horizontal lift of Zt relative to a connection on T ∗ Q × M , which is obtained by averaging the Cartan connection and is called the Cartan–Hannay–Berry connection. The holonomy of this connection is the Hannay–Berry phase of a slowly moving constrained system. For details of this approach, see Marsden, Montgomery, and Ratiu [1990]. Exercises 8.8-1. Consider the particle in a hoop of §8.7. For this problem, identify all the elements of formula (8.8.2) and use that to obtain the Lagrangian (8.7.1). 8.8-2. Consider the particle in a rotating hoop discussed in §2.8. (a) Use the tools of this section to obtain the Lagrangian given in §2.8. (b) Suppose that the hoop rotates freely. Can you still use the tools of part(a)? If so, compute the new Lagrangian and point out the differences between the two cases. (c) Analyze, in the same fashion as in §2.8, the equilibria of the free system. Does this system also bifurcate? 8.8-3. Set up the equations for the Foucault pendulum using the ideas in this section. 8.8-4. Consider again the mechanical system in Exercise 2.8-6, but this time hang a spherical pendulum from the rotating arm. Investigate the geometric phase when the arm is swung once around. (Consider doing the ⊥ experiment!) Is the term Zt 2 really small in this example? ........................... 15 July 1998—18h02 ...........................

270. 256 8. Variational Principles, Constraints, and Rotating Systems 8.9 Routh Reduction An abelian version of Lagrangian reduction was known to Routh by around 1860. A modern account was given in Arnold [1988] and, motivated by that, Marsden and Scheurle [1993a] gave a geometrization and a generalization of the Routh procedure to the nonabelian case. In this section we give an elementary classical description in preparation for more sophisticated reduction procedures, such as Euler–Poincar´ e reduction in Chapter 13. We assume that Q is a product of a manifold S and a number, say k, of copies of the circle S 1 , namely Q = S × (S 1 × · · · × S 1 ). The factor S, called shape space, has coordinates denoted x1 , . . . , xm and coordinates on the other factors are written θ1 , . . . , θk . Some or all of the factors of S 1 can be replaced by R if desired, with little change. We assume that the variables θa , a = 1, . . . , k are cyclic, that is, they do not appear explicitly in the Lagrangian, although their velocities do. As we shall see after Chapter 9 is studied, invariance of L under the action of the abelian group G = S 1 × · · · × S 1 is another way to express that fact that θa are cyclic variables. That point of view indeed leads ultimately to deeper insight, but here we focus on some basic calculations done “by hand,” in coordinates. A basic class of examples (for which Exercises 8.9-1 and 8.9-2 provide specific instances) are those for which the Lagrangian L has the form kinetic minus potential energy: ˙ ˙ 2 ˙ ˙ ˙ ˙ 2 ˙ ˙ L(x, x, θ) = 1 gαβ (x)xα xβ + gaα (x)xα θa + 1 gab (x)θa θb − V (x), (8.9.1) where there is a sum over α, β from 1 to m and over a, b from 1 to k. Even in simple examples, such as the double spherical pendulum or the simple pendulum on a cart (Exercise 8.9-2), the matrices gαβ , gaα , gab can depend on x. ˙ Because θa are cyclic, the corresponding conjugate momenta ∂L pa = (8.9.2) ˙ ∂ θa are conserved quantities. In the case of the Lagrangian (8.9.1), these momenta are given by ˙ pa = gaα xα + gab θb . ˙ Definition 8.9.1. The classical Routhian is defined by setting pa = µa = constant and performing a partial Legendre transformation in the variables θa : ˙ ˙ ˙ ˙ Rµ (x, x) = L(x, x, θ) − µa θa , (8.9.3) pa =µa ˙ where it is understood that the variable θa is eliminated using the equation pa = µa and µa is regarded as a constant. ........................... 15 July 1998—18h02 ...........................

271. 8.9 Routh Reduction 257 Now consider the Euler–Lagrange equations: d ∂L ∂L − a = 0; (8.9.4) dt ∂ xa ˙ ∂x we attempt to write these as Euler–Lagrange equations for a function from ˙ which θa has been eliminated. We claim that the Routhian Rµ does the job. To see this, we compute the Euler–Lagrange expression for Rµ using the chain rule: d ∂Rµ ∂Rµ d ∂L ˙ ∂L ∂ θa − = + dt ∂ xα ˙ ∂xα dt ∂x ˙ α ˙a ∂ xα ∂θ ˙ ∂L ˙ ∂L ∂ θa d ˙ ∂ θa ˙ ∂ θa − + − µa + µa . ∂xα ˙ ∂ θa ∂xα dt ∂ xα ˙ ∂xα The first and third terms vanish by (8.9.4) and the remaining terms vanish using µa = pa . Thus, we have proved: Proposition 8.9.2. The Euler–Lagrange equations (8.9.4) for L(x, x, θ)˙ ˙ together with the convervation laws pa = µa are equivalent to the Euler– Lagrange equations for the Routhian Rµ (x, x) together with pa = µa . ˙ The Euler–Lagrange equations for Rµ are called the reduced Euler– Lagrange equations since the configurations space Q with variables (xa , θa ) has been reduced to the configuration space S with variables xα . In what follows we shall make the following notational conventions: g ab denote the entries of the inverse matrix of the m × m matrix [gab ], and similarly, g αβ denote the entries of the inverse of the k × k matrix [gαβ ]. We will not use the entries of the inverse of the whole matrix tensor on Q, so there is no danger of confusion. Proposition 8.9.3. For L given by (8.9.1) we have Rµ (x, x) = gaα g ac µc xa + ˙ ˙ 1 2 (gαβ − gaα g ac gcβ ) xa xβ − Vµ (x), ˙ ˙ (8.9.5) where Vµ (x) = V (x) + 1 g ab µa µb 2 is the amended potential . Proof. ˙ ˙ We have µa = gaα xα + gab θb , so ˙ θa = g ab µb − g ab gbα xα . ˙ (8.9.6) Substituting this in the definition of Rµ gives Rµ (x, x) = 1 gαβ (x)xα xβ + (gaα xα ) g ac µc − g ac gcβ xβ ˙ 2 ˙ ˙ ˙ ˙ + 1 gab g ac µc − g ac gcβ xβ 2 ˙ g bd µd − g bd gdγ xγ ˙ − µa g ac µc − g ac gcβ xβ − V (x). ˙ ........................... 15 July 1998—18h02 ...........................

272. 258 8. Variational Principles, Constraints, and Rotating Systems The terms linear in x are: ˙ gaα g ac µc xα − gab g ac µc g bd gdγ xγ + µa g ac gcβ xβ = gaα g ac µc xα , ˙ ˙ ˙ ˙ while the terms quadratic in x are ˙ 1 2 (gαβ − gaα g ac gcβ )xα xβ , ˙ ˙ and the terms dependent only on x are −Vµ (x), as required. Note that Rµ has picked up a term linear in the velocity, and the potential as well as the kinetic energy matrix (the mass matrix ) have both been modified. The term linear in the velocities has the form Aa µa xα , where Aa = α ˙ α g ab gbα . The Euler–Lagrange expression for this term can be written d a ∂ ∂Aa ∂Aa β A µa − α Aa µa xβ = ˙ α − µa xβ , ˙ dt α ∂x β ∂xβ ∂xα which is denoted Bαβ µa xβ . If we think of the one form Aa dxα , then Bαβ a ˙ α a a is its exterior derivative. The quantities Aα are called connection coeffi- a cients and Bαβ are called the curvature coefficients. Introducing the modified (simpler) Routhian, obtained by deleting the terms linear in x,˙ ˜ Rµ = 1 gαβ − gaα g ab gbβ xα xβ − Vµ (x), ˙ ˙ 2 the equations take the form ˜ d ∂ Rµ ˜ ∂ Rµ α − = −Bαβ µa xβ , a ˙ (8.9.7) dt ∂ x ˙ ∂xα which is the form that makes intrinsic sense and generalizes. The extra terms have the structure of magnetic, or Coriolis, terms that we have seen in a variety of earlier contexts. The above gives a hint of the large amount of geometry hidden behind the apparently simple process of Routh reduction. In particular, connections Aa and their curvatures Bαβ play an important role in more general α a theories, such as those involving nonablelian symmetry groups (like the rotation group). Another suggestive hint of more general theories is that the kinetic term in (8.9.5) can be written in the following way gαβ gαa xβ ˙ 2 (x , −Aδ x ) ˙a 1 a δ ˙ gαa gab −Ab xγ γ˙ which also exhibits its positive definite nature. ........................... 15 July 1998—18h02 ...........................

273. 8.9 Routh Reduction 259 Routh himself (in the mid 1800’s) was very interested in rotating mechanical systems, such as those possessing an angular momentum conservation laws (see the exercises). In this context, Routh used the term “steady motion” for dynamic motions that were uniform rotations about a fixed axis. We may identify these with equlibria of the reduced Euler–Lagrange equations. Since the Coriolis term does not affect conservation of energy (we have seen this earlier with the dynamics of a particle in a magnetic field), we can apply the Lagrange–Dirichlet test to conclude that: Proposition 8.9.4 (Routh’s stability criterion). Steady motions correspond to critical points xe of the amended potential Vµ . If d2 Vµ (xe ) is positive definite, then the steady motion xe is stable. When more general symmetry groups are involved, one speaks of relative equlilibria rather than steady motions, a change of terminology due to Poincar´ around 1890. This is the beginning of a more sophisticated theory e of stability, leading up to the energy-momentum method outlined in §1.7. Exercises 8.9-1. Carry out Routh reduction for the spherical pendulum 8.9-2. Carry out Routh reduction for the planar pendulum on a cart, as in Figure 8.9.1 m θ g l = pendulum length l m = pendulum bob mass M = cart mass g = acceleration due to gravity M s Figure 8.9.1. A pendulum on a cart. 8.9-3 (Two-body problem). Compute the amended potential for the planar motion of a particle moving in a central potential V (r). Compare the result with the “effective potential” found in, for example, Goldstein [1980]. ........................... 15 July 1998—18h02 ...........................

274. 260 8. Variational Principles, Constraints, and Rotating Systems 8.9-4. Let L be a Lagrangian on T Q and let ˆ Rµ (q, q) = L(q, q) + Aa µa q a , ˙ ˙ α where Aa is an Rk -valued one-form on T Q and µ ∈ Rk∗ . (a) Write Hamilton’s principle for L as a Lagrange–D’Alembert principle ˆ for Rµ . ˆ ˆ (b) Letting H µ be the Hamiltonian associated with Rµ , show that the original Euler–Lagrange equations for L can be written as ˆ ∂H µ qα = ˙ ∂pα ˆ ∂H µ ˆ ∂H µ a pα = ˙ + βαβ µb ∂q α ∂pβ ........................... 15 July 1998—18h02 ...........................

275. This is page 261 Printer: Opaque this 9 An Introduction to Lie Groups To prepare for the next chapters, we present some basic facts about Lie groups. Alternative expositions and additional details can be obtained from Abraham and Marsden [1978], Olver [1986], and Sattinger and Weaver [1986]. In particular, in this book we shall require only elementary facts about the general theory and a knowledge of a few of the more basic groups, such as the rotation and Euclidean groups. Here are how some of the basic groups arise in mechanics: Linear and Angular Momentum. These arise as conserved quantities associated with the groups of translations and rotations in space. Rigid Body. Consider a free rigid body rotating about a its center of mass, taken to be the origin. “Free” means that there are no external forces, and “rigid” means that the distance between any two points of the body is unchanged during the motion. Consider a point X of the body at time t = 0, and denote its position at time t by f (X, t). Rigidity of the body and the assumption of a smooth motion imply that f (X, t) = A(t)X, where A(t) is a proper rotation, that is, A(t) ∈ SO(3), the proper rotation group of R3 , the 3 × 3 orthogonal matrices with determinant 1. The set SO(3) will be shown to be a three-dimensional Lie group and, since it describes any possible position of the body, it serves as the conﬁguration space. The group SO(3) also plays a dual role of a symmetry group since the same physical motion is described if we rotate our coordinate axes. Used as a symmetry group, SO(3) leads to conservation of angular momentum.

276. 262 9. An Introduction to Lie Groups Heavy Top. Consider a rigid body moving with a fixed point but under the influence of gravity. This problem still has a configuration space SO(3), but the symmetry group is only the circle group S 1 , consisting of rotations about the direction of gravity. One says that gravity has broken the symmetry from SO(3) to S 1 . This time, “eliminating” the S 1 symmetry “mys- teriously” leads one to the larger Euclidean group SE(3) of rigid motion of R3 . This is a manifestation of the general theory of semidirect products (see the Introduction, where we showed that the heavy top equations are Lie–Poisson for SE(3), and Marsden, Ratiu, and Weinstein [1984a,b]). Incompressible Fluids. Let Ω be a region in R3 that is filled with a moving incompressible fluid, and is free of external forces. Denote by η(X, t) the trajectory of a fluid particle which at time t = 0 is at X ∈ Ω. For fixed t the map ηt defined by ηt (X) = η(X, t) is a diffeomorphism of Ω. In fact, since the fluid is incompressible, we have ηt ∈ Diff vol (Ω), the group of volume-preserving diffeomorphisms of Ω. Thus, the configuration space for the problem is the infinite-dimensional Lie group Diff vol (Ω). Using Diff vol (Ω) as a symmetry group leads to Kelvin’s circulation theorem as a conservation law. See Marsden and Weinstein [1983]. Compressible Fluids. In this case the configuration space is the whole diffeomorphism group Diff(Ω). The symmetry group consists of density- preserving diffeomorphisms Diff ρ (Ω). The density plays a role similar to that of gravity in the heavy top and again leads to semidirect products, as does the next example. Magnetohydrodynamics (MHD). This example is that of a compressible fluid consisting of charged particles with the dominant electromagnetic force being the magnetic field produced by the particles themselves (possibly together with an external field). The configuration space remains Diff(Ω) but the fluid motion is coupled with the magnetic field (regarded as a two-form on Ω). Maxwell-Vlasov Equations. Let f (x, v, t) denote the density function of a collisionless plasma. The function f evolves in time by means of a time-dependent canonical transformation on R6 , that is, (x, v)-space. In ∗ other words, the evolution of f can be described by ft = ηt f0 where f0 is the initial value of f , ft its value at time t, and ηt is a canonical transformation. Thus, Diff can (R6 ), the group of canonical transformations plays an important role. Maxwell’s Equations Maxwell’s equations for electrodynamics are invariant under gauge transformations that transform the magnetic (or 4) potential by A → A + ϕ. This gauge group is an infinite-dimensional Lie group. The conserved quantity associated with the gauge symmetry in this case is the charge. ........................... 15 July 1998—18h02 ...........................

277. 9.1 Basic Definitions and Properties 263 9.1 Basic Definitions and Properties Definition 9.1.1. A Lie group is a (Banach) manifold G that has a group structure consistent with its manifold structure in the sense that group multiplication µ : G × G → G; (g, h) → gh is a C ∞ map. The maps Lg : G → G; h → gh, and Rh : G → G; g → gh are called the left and right translation maps. Note that Lg1 ◦ Lg2 = Lg1 g2 and Rh1 ◦ Rh2 = Rh2 h1 . If e ∈ G denotes the identity element, then Le = Id = Re and so (Lg )−1 = Lg−1 and (Rh )−1 = Rh−1 . Thus, Lg and Rh are diffeomorphisms for each g and h. Notice that Lg ◦ Rh = Rh ◦ Lg , that is, left and right translation commute. By the chain rule, Tgh Lg−1 ◦ Th Lg = Th (Lg−1 ◦ Lg ) = Id . Thus, Th Lg is invertible. Likewise, Tg Rh is an isomorphism. We now show that the inversion map I : G → G; g → g −1 is C ∞ . Indeed, consider solving µ(g, h) = e for h as a function of g. The partial derivative with respect to h is just Th Lg , which is an isomorphism. Thus, the solution g −1 is a smooth function of g by the implicit function theorem. Lie groups can be finite- or infinite-dimensional. For a first reading of this section, the reader may wish to assume G is finite dimensional.1 Examples (a) Any Banach space V is an Abelian Lie group with group operations µ : V × V → V, µ(x, y) = x + y, and I : V → V, I(x) = −x. The identity is just the zero vector. We call such a Lie group a vector group. 1 We caution that some interesting infinite-dimensional groups (such as groups of diffeomorphisms) are not Banach–Lie groups in the (naive) sense just given. ........................... 15 July 1998—18h02 ...........................

278. 264 9. An Introduction to Lie Groups (b) The group of linear isomorphisms of Rn to Rn is a Lie group of dimension n2 , called the general linear group and denoted GL(n, R). It is a smooth manifold, since it is an open subset of the vector space L(Rn , Rn ) of all linear maps of Rn to Rn . Indeed, GL(n, R) is the inverse image of R{0} under the continuous map A → det A of L(Rn , Rn ) to R. For A, B ∈ GL(n, R), the group operation is composition µ : GL(n, R) × GL(n, R) → GL(n, R) given by (A, B) → A ◦ B, and the inversion map is I : GL(n, R) → GL(n, R), defined by I(A) = A−1 . Group multiplication is the restriction of the continuous bilinear map (A, B) ∈ L(Rn , Rn ) × L(Rn , Rn ) → A ◦ B ∈ L(Rn , Rn ). Thus, µ is C ∞ and so GL(n, R) is a Lie group. The group identity element e is the identity map on Rn . If we choose a basis in Rn , we can represent each A ∈ GL(n, R) by an invertible (n × n)- matrix. The group operation is then matrix multiplication µ(A, B) = AB and I(A) = A−1 is matrix inversion. The identity element e is the n × n identity matrix. The group operations are obviously smooth since the formulas for the product and inverse of matrices are smooth (rational) functions of the matrix components. (c) In the same way, one sees that for a Banach space V , the group, GL(V, V ), of invertible elements of L(V, V ) is a Banach Lie group. For the proof that this is open in L(V, V ), see Abraham, Marsden, and Ratiu [1988]. Further examples are given in the next section. Charts. Given any local chart on G, one can construct an entire atlas on the Lie group G by use of left (or right) translations. Suppose, for example, that (U, ϕ) is a chart about e ∈ G, and that ϕ : U → V . Define a chart (Ug , ϕg ) about g ∈ G by letting Ug = Lg (U ) = {Lg h | h ∈ U } and defining ϕg = ϕ ◦ Lg−1 : Ug → V, h → ϕ(g −1 h). The set of charts {(Ug , ϕg )} forms an atlas provided one can show that the transition maps ϕg1 ◦ ϕ−1 = ϕ ◦ Lg−1 g2 ◦ ϕ−1 : ϕg2 (Ug1 ∩ Ug2 ) → ϕg1 (Ug1 ∩ Ug2 ) g2 1 ........................... 15 July 1998—18h02 ...........................

279. 9.1 Basic Definitions and Properties 265 are diffeomorphisms (between open sets in a Banach space). But this follows from the smoothness of group multiplication and inversion. Invariant Vector Fields. A vector field X on G is called left invariant if for every g ∈ G, L∗ X = X, that is, if g (Th Lg )X(h) = X(gh) for every h ∈ G. We have the commutative diagram in Figure 9.1.1 and illustrate the geometry in Figure 9.1.2. T Lg E TG TG T T X X G E G Lg Figure 9.1.1. The commutative diagram for a left invariant vector field. ThLg X(gh) X(h) h gh Figure 9.1.2. A left invariant vector field. Let XL (G) denote the set of left invariant vector fields on G. If g ∈ G, and X, Y ∈ XL (G) then L∗ [X, Y ] = [L∗ X, L∗ Y ] = [X, Y ], g g g so [X, Y ] ∈ XL (G). Therefore, XL (G) is a Lie subalgebra of X(G), the set of all vector fields on G. For each ξ ∈ Te G, we define a vector field Xξ on G by letting Xξ (g) = Te Lg (ξ). ........................... 15 July 1998—18h02 ...........................

280. 266 9. An Introduction to Lie Groups Then Xξ (gh) = Te Lgh (ξ) = Te (Lg ◦ Lh )(ξ) = Th Lg (Te Lh (ξ)) = Th Lg (Xξ (h)), which shows that Xξ is left invariant. The linear maps ζ1 : XL (G) → Te G, X → X(e) and ζ2 : Te G → XL (G), ξ → Xξ satisfy ζ1 ◦ ζ2 = idTe G and ζ2 ◦ ζ1 = idXL (G) . Therefore, XL (G) and Te G are isomorphic as vector spaces. The Lie Algebra of a Lie Group. Define the Lie bracket in Te G by [ξ, η] := [Xξ , Xη ](e), where ξ, η ∈ Te G and where [Xξ , Xη ] is the Jacobi–Lie bracket of vector fields. This clearly makes Te G into a Lie algebra. (Lie algebras were defined in the Introduction.) We say that this defines a bracket in Te G via left- extension. Note that by construction, [Xξ , Xη ] = X[ξ,η] , for all ξ, η ∈ Te G. Definition 9.1.2. The vector space Te G with this Lie algebra structure is called the Lie algebra of G and is denoted by g. Defining the set XR (G) of right invariant vector fields on G in the analogous way, we get a vector space isomorphism ξ → Yξ , where Yξ (g) = (Te Rg )(ξ), between Te G = g and XR (G). In this way, each ξ ∈ g defines an element Yξ ∈ XR (G), and also an element Xξ ∈ XL (G). We will prove that a relation between Xξ and Yξ is given by I∗ Xξ = −Yξ (9.1.1) where I : G → G is the inversion map: I(g) = g −1 . Since I is a diffeomorphism, (9.1.1) shows that I∗ : XL (G) → XR (G) is a vector space isomorphism. To prove (9.1.1) notice first that for u ∈ Tg G and v ∈ Th G, the derivative of the multiplication map has the expression T(g,h) µ(u, v) = Th Lg (v) + Tg Rh (u). (9.1.2) In addition, differentiating the map g → µ(g, I(g)) = e gives T(g,g−1 ) µ(u, Tg I(u)) = 0, ........................... 15 July 1998—18h02 ...........................

281. 9.1 Basic Definitions and Properties 267 for all u ∈ Tg G. This and (9.1.2) yields Tg I(u) = −(Te Rg−1 ◦ Tg Lg−1 )(u), (9.1.3) for all u ∈ Tg G. Consequently, if ξ ∈ g, and g ∈ G, we have (I∗ Xξ )(g) = (T I ◦ Xξ ◦ I −1 )(g) = Tg−1 I(Xξ (g −1 )) = −(Te Rg ◦ Tg−1 Lg )(Xξ (g −1 )) (by (9.1.3)) −1 = −Te Rg (ξ) = −Yξ (g) (since Xξ (g ) = Te Lg−1 (ξ)) and (9.1.1) is proved. Hence for ξ, η ∈ g, −Y[ξ,η] = I∗ X[ξ,η] = I∗ [Xξ , Xη ] = [I∗ Xξ , I∗ Xη ] = [−Yξ , −Yη ] = [Yξ , Yη ], so that −[Yξ , Yη ](e) = Y[ξ,η] (e) = [ξ, η] = [Xξ , Xη ](e). Therefore, the Lie algebra bracket [ , ]R in g defined by right extension of elements in g: [ξ, η]R := [Yξ , Yη ](e) is the negative of the one defined by left extension, that is, [ξ, η]R := −[ξ, η]. Examples ∼ (a) For a vector group V , Te V = V ; it is easy to see that the left invariant vector field defined by u ∈ Te V is the constant vector field: Xu (v) = u, for all v ∈ V . Therefore, the Lie algebra of a vector group V is V itself, with the trivial bracket [v, w] = 0, for all v, w ∈ V . We say that the Lie algebra is Abelian in this case. (b) The Lie algebra of GL(n, R) is L(Rn , Rn ), also denoted by gl(n), the vector space of all linear transformations of Rn , with the commutator bracket [A, B] = AB − BA. To see this, we recall that GL(n, R) is open in L(Rn , Rn ) and so the Lie algebra, as a vector space, is L(Rn , Rn ). To compute the bracket, note that for any ξ ∈ L(Rn , Rn ), Xξ : GL(n, R) → L(Rn , Rn ) given by A → Aξ, is a left invariant vector field on GL(n, R), because for every B ∈ GL(n, R), the map LB : GL(n, R) → GL(n, R) ........................... 15 July 1998—18h02 ...........................

282. 268 9. An Introduction to Lie Groups defined by LB (A) = BA is a linear mapping, and hence Xξ (LB A) = BAξ = TA LB Xξ (A). Therefore, by the local formula [X, Y ](x) = DY (x) · X(x) − DX(x) · Y (x), we get [ξ, η] = [Xξ , Xη ](I) = DXη (I) · Xξ (I) − DXξ (I) · Xη (I). But Xη (A) = Aη is linear in A, so DXη (I) · B = Bη. Hence DXη (I) · Xξ (I) = ξη, and similarly DXξ (I) · Xη (I) = ηξ. Thus, L(R , R ) has the bracket n n [ξ, η] = ξη − ηξ. (9.1.4) (c) We can also establish (9.1.4) by a coordinate calculation. Choosing a basis on Rn , each A ∈ GL(n, R) is specified by its components Ai such j that (Av)i = Ai v j (sum on j ). Thus, a vector field X on GL(n, R) has j i i the form X(A) = i,j Cj (A)(∂/∂Aj ). It is checked to be left invariant i provided there is a matrix (ξj ) such that for all A, ∂ X(A) = Ai ξj k k . i,j ∂Aij i k i If Y (A) = i,j Ak ηj (∂/∂Aj ) is another left invariant vector field, we have ∂ ∂f (XY )[f ] = Ai ξj k k Al ηp m m ∂Aij ∂Alp ∂f = Ai ξj δi δm ηp k k l j m + (second derivatives) ∂Alp ∂f = Ai ξj ηm k k j + (second derivatives), ∂Ai j where we used ∂As /∂Ak = δs δm . Therefore, the bracket is the left invari- m j k j ant vector field [X, Y ] given by ∂f [X, Y ][f ] = (XY − Y X)[f ] = Ai (ξj ηm − ηj ξm ) k k j k j . ∂Aim This shows that the vector field bracket is the usual commutator bracket of (n × n)-matrices, as before. ........................... 15 July 1998—18h02 ...........................

283. 9.1 Basic Definitions and Properties 269 One-parameter Subgroups and the Exponential Map. If Xξ is the left invariant vector field corresponding to ξ ∈ g, there is a unique integral curve γξ : R → G of Xξ starting at e; γξ (0) = e and γξ (t) = Xξ (γξ (t)). We claim that γξ (s + t) = γξ (s)γξ (t), which means that γξ (t) is a smooth one-parameter subgroup. Indeed, as functions of t, both sides equal γξ (s) at t = 0 and both satisfy the differential equation σ (t) = Xξ (σ(t)) by left invariance of Xξ , so they are equal. Left invariance or γξ (t + s) = γξ (t)γξ (s) also shows that γξ (t) is defined for all t ∈ R. Definition 9.1.3. The exponential map exp : g → G is defined by exp(ξ) = γξ (1). We claim that exp(sξ) = γξ (s). Indeed, for fixed s ∈ R, the curve t → γξ (ts) which at t = 0 passes through e, satisfies the differential equation d γξ (ts) = sXξ (γξ (ts)) = Xsξ (γξ (ts)). dt Since γsξ (t) satisfies the same differential equation and passes through e at t = 0, it follows that γsξ (t) = γξ (ts). Putting t = 1 yields exp(sξ) = γξ (s). Hence the exponential mapping maps the line sξ in g onto the one- parameter subgroup γξ (s) of G, which is tangent to ξ at e. It follows from left invariance that the flow Ftξ of Xξ satisfies Ftξ (g) = gFtξ (e) = gγξ (t), so Ftξ (g) = g exp(tξ) = Rexp tξ g. Let γ(t) be a smooth one-parameter subgroup of G, so γ(0) = e in particular. We claim that γ = γξ , where ξ = γ (0). Indeed, taking the derivative at s = 0 in the relation γ(t + s) = γ(t)γ(s) gives dγ(t) d = Lγ(t) γ(s) = Te Lγ(t) γ (0) = Xξ (γ(t)), dt ds s=0 so that γ = γξ since both equal e at t = 0. In other words, all smooth one-parameter subgroups of G are of the form exp tξ for some ξ ∈ g. Since everything proved above for Xξ can be repeated for Yξ , it follows that the exponential map is the same for the left and right Lie algebras of a Lie group. From smoothness of the group operations and smoothness of the solutions of differential equations with respect to initial conditions, it follows ........................... 15 July 1998—18h02 ...........................

284. 270 9. An Introduction to Lie Groups that exp is a C ∞ map. Differentiating the identity exp(sξ) = γξ (s) with respect to s at s = 0 shows that T0 exp = idg . Therefore, by the inverse function theorem, exp is a local diffeomorphism from a neighborhood of zero in g onto a neighborhood of e in G. In other words, the exponential map defines a local chart for G at e; in finite dimensions, the coordinates associated to this chart are called the canonical coordinates of G. By left translation, this chart provides an atlas for G. (For typical infinite- dimensional groups like diffeomorphism groups, exp is not locally onto a neighborhood of the identity. It is also not true that the exponential map is a local diffeomorphism at any ξ = 0, even for finite-dimensional Lie groups.) It turns out that the exponential map characterizes not only the smooth one-parameter subgroups of G, but the continuous ones as well, as given in the next Proposition. Tudor: give a reference, Proposition 9.1.4. Let r : R → G be a continuous one-parameter sub- put proof on group of G. Then r is automatically smooth and hence r(t) = exp tξ, for internet or some ξ ∈ g. leave out. Examples (a) Let G = V be a vector group, that is, V is a vector space and the group operation is vector addition. Then g = V and exp : V → V is the identity mapping. (b) Let G = GL(n, R); so g = L(Rn , Rn ). For every A ∈ L(Rn , Rn ), the mapping γA : R → GL(n, R) defined by ∞ ti i t→ A i=0 i! is a one-parameter subgroup, because γA (0) = I and ∞ ti−1 γA (t) = Ai = γA (t)A. i=0 (i − 1)! Therefore, the exponential mapping is given by ∞ Ai exp : L(Rn , Rn ) → GL(n, Rn ), A → γA (1) = . i=0 i! As is customary, we will write ∞ Ai eA = . i=0 i! We sometimes write expG : g → G when there is more than one group involved. ........................... 15 July 1998—18h02 ...........................

285. 9.1 Basic Definitions and Properties 271 (c) Let G1 and G2 be Lie groups with Lie algebras g1 and g2 . Then G1 × G2 is a Lie group with Lie algebra g1 × g2 , and the exponential map is given by exp : g1 × g2 → G1 × G2 ; (ξ1 , ξ2 ) → (exp1 (ξ1 ), exp2 (ξ2 )). Computing Brackets. Here is a computationally useful formula for the bracket. One follows these three steps: 1. Calculate the inner automorphisms Ig : G → G, where Ig (h) = ghg −1 . 2. Differentiate Ig (h) with respect to h at h = e to produce the adjoint operators Adg : g → g; Adg ·η = Te Ig · η. Note that (see Figure 9.1.3); Adg η = Tg−1 Lg · Te Rg−1 · η. 3. Differentiate Adg η with respect to g at e in the direction ξ to get [ξ, η], that is, Te ϕη · ξ = [ξ, η], (9.1.5) where ϕη (g) = Adg η. TeLg Adg g e TgRg–1 Figure 9.1.3. The adjoint mapping is the linearization of conjugation. Proposition 9.1.5. Formula (9.1.5) is valid. ........................... 15 July 1998—18h02 ...........................

286. 272 9. An Introduction to Lie Groups Proof. Denote by ϕt (g) = g exp tξ = Rexp tξ g, the ﬂow of Xξ . Then d [ξ, η] = [Xξ , Xη ](e) = Tϕ (e) ϕ−1 · Xη (ϕt (e)) t dt t t=0 d = Texp tξ Rexp(−tξ) Xη (exp tξ) dt t=0 d = Texp tξ Rexp(−tξ) Te Lexp tξ η dt t=0 d = Te (Lexp tξ ◦ Rexp(−tξ) )η dt t=0 d = Adexp tξ η , dt t=0 which is (9.1.5). Another way of expressing (9.1.5) is d d [ξ, η] = g(t)h(s)g(t)−1 , (9.1.6) dt ds s=0,t=0 where g(t) and h(s) are curves in G with g(0) = e, h(0) = e, and where g (0) = ξ and h (0) = η. Example. Consider the group GL(n, R). Formula (9.1.4) also follows from (9.1.5). Here, IA B = ABA−1 and so AdA η = AηA−1 . Diﬀerentiating this with respect to A at A = Identity in the direction ξ gives [ξ, η] = ξη − ηξ. Group Homomorphisms. Some simple facts about Lie group homomorphisms will prove useful. Proposition 9.1.6. Let G and H be Lie groups with Lie algebras g and h. Let f : G → H be a smooth homomorphism of Lie groups, that is, f (gh) = f (g)f (h), for all g, h ∈ G. Then Te f : g → h is a Lie algebra homomorphism, that is, (Te f )[ξ, η] = [Te f (ξ), Te f (η)], for all ξ, η ∈ g. In addition, f ◦ expG = expH ◦Te f. ........................... 15 July 1998—18h02 ...........................

287. 9.1 Basic Deﬁnitions and Properties 273 Proof. Since f is a group homomorphism, f ◦ Lg = Lf (g) ◦ f . Thus, T f ◦ T Lg = T Lf (g) ◦ T f from which it follows that XTe f (ξ) (f (g)) = Tg f (Xξ (g)), that is, Xξ and XTe f (ξ) are f -related . It follows that the vector ﬁelds [Xξ , Xη ] and [XTe f (ξ) , XTe f (η) ] are also f -related for all ξ, η ∈ g (see Abra- ham, Marsden, and Ratiu [1986], §4.2). Hence Te f ([ξ, η]) = (T f ◦ [Xξ , Xη ])(e) (where e = eG ) = [XTe f (ξ) , XTe f (η) ](¯) e (where e = eH = f (e)) ¯ = [Te f (ξ), Te f (η)]. Thus, Te f is a Lie algebra homomorphism. Fixing ξ ∈ g, note that α : t → f (expG (tξ)) and β : t → expH (tTe f (ξ)) are one-parameter subgroups of H. Moreover, α (0) = Te f (ξ) = β (0), and so α = β. In particular, f (expG (ξ)) = expH (Te f (ξ)), for all ξ ∈ g. Example. Proposition 9.1.5 applied to the determinant map gives the identity det(exp A) = exp(trace A) for A ∈ GL(n, R). Corollary 9.1.7. Assume that f1 , f2 : G → H are homomorphisms of Lie groups and that G is connected. If Te f1 = Te f2 , then f1 = f2 . This follows from Proposition 9.1.5 since a connected Lie group G is generated by a neighborhood of the identity element. This latter fact may be proved following these steps: 1. Show that any open subgroup of a Lie group is closed (since its complement is a union of sets homeomorphic to it). 2. Show that a subgroup of a Lie group is open if and only if it contains a neighborhood of the identity element. 3. Conclude that a Lie group is connected if and only if it is generated by arbitrarily small neighborhoods of the identity element. From Proposition 9.1.5 and the fact that the inner automorphisms are group homomorphisms, we get Corollary 9.1.8. (i) exp(Adg ξ) = g(exp ξ)g −1 , for every ξ ∈ g and g ∈ G; and (ii) Adg [ξ, η] = [Adg ξ, Adg η]. ........................... 15 July 1998—18h02 ...........................

288. 274 9. An Introduction to Lie Groups More Automatic Smoothness Results. There are some interesting results related in spirit to Proposition 9.1.4 and the preceding discussions. A striking example of this is: Theorem 9.1.9. Any continuous homomorphism of finite dimensional Lie groups is smooth. There is a remarkable consequence of this theorem. If G is a topological group (that is, the multiplication and inversion maps are continuous) one could, in principle, have more than one differentiable manifold structure making G into two non-isomorphic Lie groups (i.e., the manifold structures are not diffeomorphic) but both inducing the same topological structure. This phenomenon of “exotic structures” occurs for general manifolds. How- ever, in view of the theorem above, this cannot happen in the case of Lie groups. Indeed, since the identity map is a homeomorphism, it must be a diffeomorphism. Thus, a toplological group that is locally Euclidean, (i.e., there is an open neighborhood of the identity homeomorphic to an open ball in Rn ), admits at most one smooth manifold structure relative to which it is a Lie group. The existence part of this statement is Hilbert’s famous fifth problem: show that a locally Euclidean topological group admits a smooth (actually analytic) structure making it into a Lie group. The solution of this problem was achieved by Gleason and, independently, by Montgomery and Zippin in 1952; see Kaplansky [1971] for an excellent account of this proof. Abelian Lie Groups. Since any two elements of an Abelian Lie group G commute, it follows that all adjoint operators Adg , g ∈ G equal the identity. Therefore, by equation (9.1.5), The Lie algebra g is Abelian; that is, [ξ, η] = 0 for all ξ, η ∈ g. Examples (a) Any finite dimensional vector space, thought of as an Abelian group under addition, is an Abelian Lie group. The same is true in infite dimensions for any Banach space. The exponential map is the identity. (b) The unit circle in the complex plane S 1 = {z ∈ C | |z| = 1} is an Abelian Lie group under multiplication. The tangent space Te S 1 is the imaginary axis and we identify R with Te S 1 by t → 2πit. With this identification, the exponential map exp : R → S 1 is given by exp(t) = e2πit . Note that exp−1 (1) = Z. (c) The n-dimensional torus Tn = S 1 × · · · × S 1 ( n times ) is an Abelian Lie group. The exponential map exp : Rn → Tn is given by exp(t1 , . . . , tn ) = (e2πit1 , . . . , e2πitn ). ........................... 15 July 1998—18h02 ...........................

289. 9.1 Basic Definitions and Properties 275 Since S 1 = R/Z , it follows that Tn = R/Zn , the projection Rn → Tn being given by exp above. If G is a connected Lie group whose Lie algebra g is Abelian, the Lie group homomorphism g ∈ G → Adg ∈ GL(g) has induced Lie algebra homomorphism ξ ∈ g → adξ ∈ gl(g) the constant map equal to zero. Therefore, by Corollary 9.1.7, Adg = identity on G, for any g ∈ G. Apply Corollary 9.1.7 again, this time to the conjugation by g on G (whose induced Lie algebra homomorphism is Adg ), to conclude that it equals the identity map on G. Thus, g commutes with all elements of G; since g was arbitrary we conclude that G is Abelian. We summarize these observations in the following proposition. Proposition 9.1.10. If G is an Abelian Lie group. its Lie algebra g is also Abelian. Conversely, if G is connected and g is Abelian, then G is Abelian. The main structure theorem for Abelian Lie groups is the following. Tudor, reference Theorem 9.1.11. Every connected Abelian n-dimensional Lie group G neeed is isomorphic to a cylinder, that is, to Tk × Rn−k for some k = 1, . . . , n. Lie Subgroups. It is natural to synthesize the subgroup and submanifold concepts. Definition 9.1.12. A Lie subgroup H of a Lie group G is a subgroup of G which is also an injectively immersed submanifold of G. If H is a submanifold of G, then H is called a regular Lie subgroup. For example, the one-parameter subgroups of the torus T2 that wind densely on the torus are Lie subgroups that are not regular. The Lie algebras g and h of G and a Lie subgroup H, respectively, are related in the following way: Proposition 9.1.13. Let H be a Lie subgroup of G. Then h is a Lie subalgebra of g. Moreover, h = {ξ ∈ g | exp tξ ∈ H, for all t ∈ R}. Proof. The first statement is a consequence of Proposition 9.1.5, which also shows that exp tξ ∈ H, for all ξ ∈ h and t ∈ R. Conversely, if exp tξ ∈ H, for all t ∈ R, we have, d exp tξ ∈h dt t=0 since H is a Lie subgroup; but this equals ξ by definition of the exponential map. ........................... 15 July 1998—18h02 ...........................

290. 276 9. An Introduction to Lie Groups The following is a powerful theorem often used to find Lie subgroups. Theorem 9.1.14. If H is a closed subgroup of a Lie group G, then H is a regular Lie subgroup. Conversely, if H is a regular Lie subgroup of G then H is closed. Margin: We remind the reader that the Lie algebras appropriate to fluid dynamics Reference and plasma physics are infinite dimensional. Nevertheless, there is still, neeed with the appropriate technical conditions, a correspondence between Lie groups and Lie algebras, analogous to the preceding theorems. The reader should be warned, however, that these theorems do not naively generalize to the infinite-dimensional situation and to prove them for special cases, specialized analytical theorems may be required. The next result is sometimes called “Lie’s third fundamental theorem.” Margin: Reference Theorem 9.1.15. Let G be a Lie group with Lie algebra g, and let h be neeed a Lie subalgebra of g. Then there exists a unique connected Lie subgroup H of G whose Lie algebra is h. Quotients. If H is a closed subgroup of G, we denote by G/H, the set of left cosets, that is, the collection {gH | g ∈ G}. Let π : G → G/H be the projection g → gH. Theorem 9.1.16. There is a unique manifold structure on G/H such that the projection π : G → G/H is a smooth surjective submersion.2 Margin: The Maurer–Cartan Equations. We close this section with a proof Reference of the Maurer–Cartan structure equations on a Lie group G. Define neeed λ, ρ ∈ Ω1 (G; g), the space of g-valued one-forms on G, by λ(ug ) = Tg Lg−1 (ug ), ρ(ug ) = Tg Rg−1 (ug ). Thus, λ and ρ are Lie algebra valued one-forms on G that are defined by left and right translation to the identity respectively. Define the two-form [λ, λ] by [λ, λ](u, v) = [λ(u), λ(v)], and similarly for [ρ, ρ]. Theorem 9.1.17 (Maurer–Cartan Structure Equations). dλ + [λ, λ] = 0, dρ − [ρ, ρ] = 0. Proof. We use identity 6 from the table in §4.4. Let X, Y ∈ X(G) and let, for fixed g ∈ G, ξ = Tg Lg−1 (X(g)) and η = Tg Lg−1 (Y (g)). Thus, (dλ)(Xξ , Xη ) = Xξ [λ(Xη )] − Xη [λ(Xξ )] − λ([Xξ , Xη ]). 2A smooth map is called a submersion when its derivative is surjective. ........................... 15 July 1998—18h02 ...........................

291. 9.1 Basic Definitions and Properties 277 Since λ(Xη )(h) = Th Lh−1 (Xη (h)) = η is constant, the first term vanishes. Similarly, the second term vanishes. The third term equals λ([Xξ , Xη ]) = λ(X[ξ,η] ) = [ξ, η], and hence (dλ)(Xξ , Xη ) = −[ξ, η]. Therefore, (dλ + [λ, λ]) (Xξ , Xη ) = −[ξ, η] + [λ, λ](Xξ , Xη ) = −[ξ, η] + [λ(Xξ ), λ(Xη )] = −[ξ, η] + [ξ, η] = 0. This proves that (dλ + [λ, λ]) (X, Y )(g) = 0. Since g ∈ G was arbitrary as well as X and Y , it follows that dλ+[λ, λ] = 0. The second relation is proved in the same way but working with the right invariant vector fields Yξ , Yη . The sign in front of the second term changes since [Yξ , Yη ] = Y−[ξ,η] . Remark. If α is a (0, k)-tensor with values in a Banach space E1 , and β is a (0, l)-tensor with values in a Banach space E2 , and if B : E1 × E2 → E3 is a bilinear map, then replacing multiplication in (4.2.1) by B, the same formula defines an E3 -valued (0, k + l)-tensor on M . Therefore, using Definitions (4.2.2)–(4.2.4) if α ∈ Ωk (M, E1 ) and β ∈ Ωl (M, E2 ), then (k + l)! A(α ⊗ β) ∈ Ωk+l (M, E3 ). k!l! We shall call this expression the wedge product associated to B and denote it either by α ∧B β or B ∧ (α, β). In particular, if E1 = E2 = E3 = g and B = [ , ] is the Lie algebra bracket, then for α, β ∈ Ω1 (M ; g), we have [α, β]∧ (u, v) = [α(u), β(v)] − [α(v), β(u)] = −[β, α]∧ (u, v) for any vectors u, v tangent to M . Thus, alternatively, one can write the structure equations as dλ + 1 [λ, λ]∧ = 0, 2 dρ − 1 [ρ, ρ]∧ = 0. 2 ........................... 15 July 1998—18h02 ...........................

292. 278 9. An Introduction to Lie Groups Haar measure. One can characterize Lebesgue measure up to a multiplicative constant on Rn by its invariance under translations. Similarly, on a locally compact group there is a unique (up to a nonzero multiplicative constant) left-invariant measure, called Haar measure. For Lie groups the existence of such measures is especially simple. Proposition 9.1.18. Let G be a Lie group. Then there is a volume form µ, unique up to nonzero multiplicative constants, which is left invariant. If G is compact, µ is right invariant as well. Proof. Pick any n-form µe on Te G that is nonzero and define an n-form on Tg G by µg (v1 , . . . , vn ) = µe · (T Lg−1 v1 , . . . , T Lg−1 · vn ). Then µg is left invariant and smooth. For n = dim G, µe is unique up to a scalar factor, so µg is as well. ∗ Fix g0 ∈ G and consider Rg0 µ = cµ for a constant c. If G is compact, this relationship may be integrated, and by the change of variables formula we deduce that c = 1. Hence, µ is also right invariant. Exercises 9.1-1. Verify Adg [ξ, η] = [Adg ξ, Adg η] directly for GL(n). 9.1-2. Let G be a Lie group with group operations µ : G × G → G and I : G → G. Show that the tangent bundle T G is also a Lie group, called the tangent group of G with group operations T µ : T G×T G → T G, T I : T G → T G. 9.1-3 (Defining a Lie group by a chart at the identity). Let G be a group and suppose that ϕ : U → V is a one-to-one map from a subset U of G containing the identity element to an open subset V in a Banach space (or Banach manifold). The following conditions are necessary and sufficient for ϕ to be a chart in a Hausdorff–Banach–Lie group structure on G: (a) The set W = {(x, y) ∈ V × V | ϕ−1 (y) ∈ U } is open in V × V and the map (x, y) ∈ W → ϕ(ϕ−1 (x)ϕ−1 (y)) ∈ V is smooth. (b) For every g ∈ G, the set Vg = ϕ(gU g −1 ∩ U ) is open in V and the map x ∈ Vg → ϕ(gϕ−1 (x)g −1 ) ∈ V is smooth. 9.1-4 (The Heisenberg group). Let (Z, Ω) be a symplectic vector space and define on H := Z × S 1 the following operation: −1 (u, exp iφ)(v, exp iψ) = u + v, exp i[φ + ψ + Ω(u, v)] . ........................... 15 July 1998—18h02 ...........................

293. 9.2 Some Classical Lie Groups 279 (a) Verify that this operation gives H the structure of a non-commutative Lie group. (b) Show that the Lie algebra of H is given by h = Z ×R with the bracket operation3 [(u, φ), (v, ψ)] = (0, 2 −1 Ω(u, v)). (c) Show that [h, [h, h]] = 0, that is, h is nilpotent, and that R lies in the center of the algebra (i.e., [h, R] = 0); one says that h is a central extension of Z. 9.2 Some Classical Lie Groups The Real General Linear Group GL(n, R). In the previous section we showed that GL(n, R) is a Lie group, that it is an open subset of the vector space of all linear maps of Rn into itself, and that its Lie algebra is gl(n, R) with the commutator bracket. Since it is open in L(Rn , Rn ) = gl(n, R), the group GL(n, R) is not compact. The determinant function det : GL(n, R) → R is smooth and maps GL(n, R) onto the two components of R{0}. Thus, GL(n, R) is not connected. Denote by GL† (n, R) = {A ∈ GL(n, R) | det(A) 0} and note that it is an open (and hence closed) subgroup of GL(n, R). If GL− (n, R) = {A ∈ GL(n, R) | det(A) 0} the map A ∈ GL† (n, R) → I0 A ∈ GL− (n, R), where I0 is the diagonal matrix all of whose entries are 1 except the (1, 1)-entry which is −1, is a diﬀeomorphism. We will show below that GL† (n, R) is connected which will prove that GL† (n, R) is the connected component of the identity in GL(n, R) and that GL(n, R) has exactly two connected components. To prove this we need a theorem from linear algebra, called the Polar De- composition Theorem. To formulate it, recall that a matrix R ∈ GL(n, R) is orthogonal if RRT = RT R = I. A matrix S ∈ gl(n, R) is called symmetric if S T = S. A symmetric matrix S is called positive deﬁnite, denoted S 0, if Sv, v 0 for all v ∈ Rn , v = 0. Note that S 0 implies that S is invertible. 3 This formula for the bracket, when applied to the space Z = R 2n of the usual p’s and q’s , shows that this algebra is the same as that encountered in elementary quantum mechanics via the Heisenberg commutation relations. Hence the name “Heisenberg group.” ........................... 15 July 1998—18h02 ...........................

294. 280 9. An Introduction to Lie Groups Proposition 9.2.1 (Real Polar Decomposition Theorem). For any A ∈ GL(n, R) there exists a unique orthogonal matrix R and positive definite matrices S1 , S2 , such that A = RS1 = S2 R. (9.2.1) Proof. Recall first that any positive definite symmetric matrix has a unique square root: if λ1 , . . . , λn 0 are the eigenvalues of AT A, diagonalize AT A by writing AT A = B diag(λ1 , . . . , λn )B −1 , and then √ AT A = B diag( λ1 , . . . , λn )B −1 . √ −1 Then let S1 = AT A, which is positive definite. Define R = AS1 and note that −1 −1 −1 2 −1 RT R = S1 AT AS1 = S1 P1 S1 = I 2 since S1 = AT A by definition. Since both A and S1 are invertible, it follows that R is invertible and hence RT = R−1 , so R is an orthogonal matrix. Let us prove uniqueness of the decomposition. Let A = RS1 = RS1 . ˜˜ Then ˜˜ ˜2 A T A = S 1 R T R S1 = S 1 . ˜ However, the square root of a positive definite matrix is unique, so S1 = S1 , ˜ whence also R = R. √ Now define S2 = AAT and, as before, we conclude that A = S2 R for some orthogonal matrix R . We prove now that R = R. Indeed, A = S2 R = (R (R )T )S2 R = R ((R )T S2 R ) and (R )T S2 R 0. By uniqueness of the prior polar decomposition, we conclude that R = R and R T S2 R = S1 . Now we will use the Real Polar Decomposition Theorem to prove that GL† (n, R) is connected. Let A ∈ GL† (n, R) and decompose it as A = SR, with S positive definite and R an orthogonal matrix whose determinant is 1. We will prove later that all orthogonal matrices having determinant equal to 1 is a connected Lie group. Thus there is a continuous path R(t) of orthogonal matrices having determinant 1 such that R(0) = I and R(1) = R. Next, define the continuous path of symmetric matrices S(t) = I +t(S − I) and note that S(0) = I and S(1) = S. Moreover, S(t)v, v = [I + t(S − I)]v, v = v 2 + t Sv, v − t v 2 = (1 − t) v 2 + t Sv, v 0, for all t ∈ [0, 1] since Sv, v 0 by hypothesis. Thus S(t) is a continuous path of positive definite matrices connecting I to S. We conclude that ........................... 15 July 1998—18h02 ...........................

295. 9.2 Some Classical Lie Groups 281 A(t) := S(t)R(t) is a continuous path of matrices whose determinant is strictly positive connecting A(0) = S(0)R(0) = I to A(1) = S(1)R(1) = SR = A. Thus, we have proved the following: Proposition 9.2.2. The group GL(n, R) is a noncompact disconnected n2 -dimensional Lie group whose Lie algebra gl(n, R) consists of all n × n matrices with the bracket [A, B] = AB − BA. The Real Special Linear Group SL(n, R). Let det : L(Rn , Rn ) → R be the determinant map and recall that GL(n, R) = {A ∈ L(Rn , Rn ) | det A = 0}, so GL(n, R) is open in L(Rn , Rn ). Notice that R{0} is a group under multiplication and that det : GL(n, R) → R{0} is a Lie group homomorphism because det(AB) = (det A)(det B). Lemma 9.2.3. The map det : GL(n, R) → R{0} is C ∞ and its derivative is given by D detA ·B = (det A) trace(A−1 B). Proof. The smoothness of det is clear from its formula in terms of matrix elements. Using the identity det(A + λB) = (det A) det(I + λA−1 B), it suﬃces to prove d det(I + λC) = tr C. dλ λ=0 This follows from the identity for the characteristic polynomial det(I + λC) = 1 + λ tr C + · · · + λn det C. Deﬁne the real special linear group SL(n, R) by −1 SL(n, R) = {A ∈ GL(n, R) | det A = 1} = det(1). (9.2.2) ........................... 15 July 1998—18h02 ...........................

296. 282 9. An Introduction to Lie Groups From Proposition 9.1.14 it follows that SL(n, R) is a closed Lie subgroup of GL(n, R). However, this method invokes a rather subtle result to prove something that is actually straightforward. In fact, it follows from Lemma 9.2.3 that det : GL(n, R) → R is a submersion, so SL(n, R) = det−1 (1) is a smooth closed submanifold and hence a closed Lie subgroup. The tangent space to SL(n, R) at A ∈ SL(n, R) therefore consists of all matrices B such that tr(A−1 B) = 0. In particular, the tangent space at the identity consists of the matrices with trace zero. We have seen that the Lie algebra of GL(n, R) is L(Rn , Rn ) = gl(n, R) with the Lie bracket given by [A, B] = AB − BA. It follows that the Lie algebra sl(n, R) of SL(n, R) consists of the set of n × n matrices having trace zero, with the bracket [A, B] = AB − BA. Since tr(B) = 0 imposes one condition on B, it follows that dim[sl(n, R)] = n2 − 1. In dealing with classical Lie groups it is useful to introduce the following inner product on gl(n, R): A, B = trace(AB T ). (9.2.3) It is straightforward to verify all axioms of an inner product. Note also that n 2 A = a2 , ij (9.2.4) i,j=1 which shows that this norm on gl(n, R) coincides with the Euclidean norm 2 on Rn . We shall use this norm to show that SL(n, R) is not compact. Indeed, all matrices of the form   1 0 ... t 0 1 . . . 0   . . .. . . . . . . . . 0 0 ... 1 √ are elements of SL(n, R) whose norm equals n + t2 for any t ∈ R. Thus, SL(n, R) is not a bounded subset of gl(n, R) and hence is not compact. Finally, let us prove that SL(n, R) is connected. As before, we shall use the Real Polar Decompoistion Theorem and the fact, to be proved later, that all orthogonal matrices having determinant equal to 1 is a connected Lie group. If A ∈ SL(n, R) decompose it as A = SR, where R is an orthogonal matrix having determinant 1 and S is a positive deﬁnite matrix having determinant 1. Since S is symmetric, it can be diagonalized, ........................... 15 July 1998—18h02 ...........................

297. 9.2 Some Classical Lie Groups 283 that is, S = B diag(λ1 , . . . , λn )B −1 for some orthogonal matrix B and λ1 , . . . , λn 0. Define the continuous path n−1 S(t) = B diag 1 + tλ1 , . . . , 1 + tλn−1 , 1/ (1 + tλi ) i=1 for t ∈ [0, 1] and note that, by construction, det S(t) = 1, S(t) is symmetric, S(t) is positive definite since each entry 1 + tλi 0 for t ∈ [0, 1], and S(0) = I, S(1) = S. Now let R(t) be a continuous path of orthogonal matrices of determinant 1 such that R(0) = I and R(1) = R. Therefore, A(t) = S(t)R(t) is a continuous path in SL(n, R) satisfying A(0) = I and A(1) = SR = A, thereby showing that SL(n, R) is connected. Proposition 9.2.4. The Lie group SL(n, R) is a noncompact connected (n2 − 1)-dimensional Lie group whose Lie algebra sl(n, R) consists of the (n × n) matrices with trace zero (or linear maps of Rn to Rn with trace zero) with the bracket [A, B] = AB − BA. The Orthogonal Group O(n). On Rn we use the standard inner product n x, y = xi y i , i=1 where x = (x , . . . , x ) ∈ R and y = (y 1 , . . . , y n ) ∈ Rn . Recall that a 1 n n linear map A ∈ L(Rn , Rn ) is orthogonal if Ax, Ay = x, y , (9.2.5) 1/2 for all x, y ∈ R. In terms of the norm x = x, x , one sees from the polarization identity that A is orthogonal iff Ax = x , for all x ∈ Rn , or in terms of the transpose AT , which is defined by Ax, y = x, AT y , we see that A is orthogonal iff AAT = I. Let O(n) denote the orthogonal elements of L(Rn , Rn ). For A ∈ O(n), we see that 1 = det(AAT ) = (det A)(det AT ) = (det A)2 ; hence det A = ±1 and so A ∈ GL(n, R). Furthermore, if A, B ∈ O(n) then ABx, ABy = Bx, By = x, y and so AB ∈ O(n). Letting x = A−1 x and y = A−1 y, we see that x, y = Ax , Ay = x , y , that is, x, y = A−1 x, A−1 y ; ........................... 15 July 1998—18h02 ...........................

298. 284 9. An Introduction to Lie Groups hence A−1 ∈ O(n). Let S(n) denote the vector space of symmetric linear maps of Rn to itself, and let ψ : GL(n, R) → S(n) be deﬁned by ψ(A) = AAT . We claim that I is a regular value of ψ. Indeed, if A ∈ ψ −1 (I) = O(n), the derivative of ψ is Dψ(A) · B = AB T + BAT which is onto (to hit C, take B = CA/2). Thus, ψ −1 (I) = O(n) is a closed Lie subgroup of GL(n, R), called the orthogonal group. Since O(n) is closed and bounded in L(Rn , Rn ) (the norm of A ∈ O(n) is 1/2 1/2 √ A = trace(AT A) = (trace I) = n ), it is compact. We shall see in §9.3 that O(n) is not connected, but has two connected components, one where det = +1 and the other where det = −1. The Lie algebra o(n) of O(n) is ker Dψ(I), namely, the skew-symmetric linear maps with the usual commutator bracket [A, B] = AB − BA. The space of skew-symmetric n × n matrices has dimension equal to the number of entries above the diagonal, namely, n(n − 1)/2. Thus, dim[O(n)] = 1 n(n − 1). 2 The special orthogonal group is deﬁned as SO(n) = O(n) ∩ SL(n, R), that is, SO(n) = {A ∈ O(n) | det A = +1}. (9.2.6) Since SO(n) is the kernel of det : O(n) → {−1, 1}, that is, SO(n) = det−1 (1), it is an open and closed Lie subgroup of O(n), hence is compact. We shall prove in §9.3 that SO(n) is the connected component of O(n) containing the identity I, and so has the same Lie algebra as O(n). We summarize: Proposition 9.2.5. The Lie group O(n) is a compact Lie group of dimension n(n − 1)/2. Its Lie algebra o(n) is the space of skew-symmetric n × n matrices with bracket [A, B] = AB − BA. The connected component of the identity in O(n) is the compact Lie group SO(n) which has the same Lie algebra so(n) = o(n). O(n) has two connected components. the last sentence Rotations in the Plane SO(2). We parametrize begins with a symbol? S 1 = {x ∈ R2 | x = 1} by the polar angle θ, 0 ≤ θ 2π. For each θ ∈ [0, 2π], let cos θ − sin θ Aθ = , sin θ cos θ ........................... 15 July 1998—18h02 ...........................

299. 9.2 Some Classical Lie Groups 285 using the standard basis of R2 . Then Aθ ∈ SO(2) represents a counterclockwise rotation through the angle θ. Conversely, if a1 a2 A= a3 a4 is orthogonal, the relations a2 + a2 = 1, a2 + a2 = 1, 1 2 3 4 a1 a3 + a2 a4 = 0, det A = a1 a4 − a2 a3 = 1 show that A = Aθ for some θ. Thus, SO(2) can be identified with S 1 ; that is, with rotations in the plane. Rotations in Space SO(3). The Lie algebra so(3) of SO(3) may be identified with R3 as follows. We define the vector space isomorphism ˆ : R3 → so(3) called the hat map, by   0 −v3 v2 v = (v1 , v2 , v3 ) → v =  v3 ˆ 0 −v1  . (9.2.7) −v2 v1 0 Note that the identity vw = v × w ˆ characterizes this isomorphism. We get (ˆ v − vu) w = u(v × w) − v(u × w) uˆ ˆ ˆ ˆ ˆ = u × (v × w) − v × (u × w) = (u × v) × w = (u × v)ˆ · w. Thus, if we put the cross product on R3 , ˆ becomes a Lie algebra isomorphism and so we can identify so(3) with R3 with the cross product as Lie bracket. We also note that the standard dot product may be written v·w = 1 2 trace vT w = − 1 trace (ˆ w) . ˆ ˆ 2 vˆ Theorem 9.2.6 (Euler’s Theorem). Every element A ∈ SO(3), A = I, is a rotation through an angle θ about an axis w. To prove this, we use the following lemma: Lemma 9.2.7. Every A ∈ SO(3) has an eigenvalue equal to 1. ........................... 15 July 1998—18h02 ...........................

300. 286 9. An Introduction to Lie Groups Proof. The eigenvalues of A are given by roots of the third degree polynomial det(A − λI) = 0. Roots occur in conjugate pairs, so at least one is real. If λ is a real root and x is a nonzero real eigenvector, Ax = λx, so Ax 2 = x 2 and Ax 2 = |λ|2 x 2 imply λ = ±1. If all three roots are real, they are (1, 1, 1) or (1, −1, −1) since det A = 1. If there is one real and two complex conjugate roots, they are (1, ω, ω ) since det A = 1. In any case one real root must be +1. ¯ Proof of Theorem 9.2.6.. By Lemma 9.2.7, the matrix A has an eigenvector w with eigenvalue 1, say Aw = w. The line spanned by w is also invariant under A. Let P be the plane perpendicular to w; that is, P = {y | w, y = 0} . Since A is orthogonal, A(P ) = P . Let e1 , e2 be an orthogonal basis in P . Then relative to (w, e1 , e2 ), A has the matrix   1 0 0 A =  0 a1 a2  . 0 a3 a4 Since a1 a2 a3 a4 lies in SO(2), A is a rotation about the axis w by some angle. Corollary 9.2.8. Any A ∈ SO(3) can be written in some orthonormal basis as the matrix   1 0 0 A =  0 cos θ − sin θ  . 0 sin θ cos θ The inﬁnitesimal version of Euler’s theorem is the following: Proposition 9.2.9. Identifying the Lie algebra so(3) of SO(3) with the Lie algebra R3 , exp(tw) is a rotation about w by the angle t w , where w ∈ R3 . Proof. To simplify the computation, we pick an orthonormal basis (e1 , e2 , e3 ) of R3 , with e1 = w/ w . Relative to this basis, w has the matrix ˆ   0 0 0 w = w  0 0 −1  . ˆ 0 1 0 ........................... 15 July 1998—18h02 ...........................

301. 9.2 Some Classical Lie Groups 287 Let   1 0 0 c(t) =  0 cos t w − sin t w . 0 sin t w cos t w Then   0 0 0 c (t) =  0 − w sin t w − w cos t w  0 w cos t w − w sin t w ˆ ˆ = c(t)w = TI Lc(t) (w) = Xw (c(t)), ˆ ˆ where Xw is the left invariant vector ﬁeld corresponding to w. Therefore, ˆ ˆ ˆ c(t) is an integral curve of Xw ; but exp(tw) is also an integral curve of Xw . ˆ Since both agree at t = 0, exp(tw) = c(t), for all t ∈ R. But the matrix ˆ deﬁnition of c(t) expresses it as a rotation by an angle t w about the axis w. Despite Euler’s theorem, it might be good to recall now that SO(3) cannot be written as S 2 × S 1 ; see Exercise 1.2-4. Amplifying on Proposition 9.2.7, we give the following explicit formula for exp ξ, where ξ ∈ so(3), which is called Rodrigues formula:  2 sin v sin v2 exp[ˆ ] = I + v v+ 1 ˆ 2 v  v2 . ˆ (9.2.8) v 2 This formula is due to Rodgiques [1840]; see also Helgason [1978], Exercise 1, p. 249 and see Altmann [1986] for some interesting history of this formula. Proof of Rodrigues’ Formula. By (9.2.7), v w = v × (v × w) = v, w v − v 2 w. ˆ 2 (9.2.9) Consequently, we have the recurrence relations v3 = − v 2 v, ˆ ˆ v4 = − v 2 v2 , ˆ ˆ v5 = v 4 v, ˆ ˆ v6 = v 4 v2 , . . . . ˆ ˆ Splitting the exponential series in odd and even powers, v 2 v 4 v 2n exp[ˆ ] = I + I − v + − · · · + (−1)n+1 + ··· v ˆ 3! 5! (2n + 1)! 1 v 2 v 4 v n−2 + − + + · · · + (−1)n−1 + · · · v2 ˆ 2! 4! 6! (2n)! sin v 1 − cos v 2 =I+ ˆ v+ ˆ v , (9.2.10) v v 2 and so the result follows from identity 2 sin2 ( v /2) = 1 − cos v . ........................... 15 July 1998—18h02 ...........................

302. 288 9. An Introduction to Lie Groups The following alternative expression, equivalent to (9.2.8), is often useful. Set n = v/ v so that n = 1. From (9.2.9) and (9.2.10) we obtain exp[ˆ ] = I + (sin v )ˆ + (1 − cos v )[n ⊗ n − I]. v n (9.2.11) Here, n ⊗ n is the matrix whose entries are ni nj , or as a bilinear form, (n ⊗ n)(α, β) = n(α)n(β). Therefore, we obtain a rotation about the unit vector n = v/ v of magnitude v . The results (9.2.8) and (9.2.11) are useful in computational solid mechanics, along with their quaternionic counterparts. We shall return to this point below in connection with SU(2); see Whittaker [1927] and Simo and Fox [1989] for more information. We next give a topological property of SO(3). Proposition 9.2.10. The rotation group SO(3) is diffeomorphic to the real projective space RP3 . Proof. To see this, map the unit ball D in R3 to SO(3) by sending (x, y, z) to the rotation about (x, y, z) through the angle π x2 + y 2 + z 2 (and (0, 0, 0) to the identity). This mapping is clearly smooth and surjective. Its restriction to the interior of D is injective. On the boundary of D, this mapping is 2 to 1, so it induces a smooth bijective map from D, with antipodal points on the boundary identified, to SO(3). It is a straightforward exercise to show that the inverse of this map is also smooth. Thus, SO(3) is diffeomorphic with D, with antipodal points on the boundary identified. However, the mapping (x, y, z) → (x, y, z, 1 − x2 − y 2 − z 2 ) is a diffeomorphism between D, with antipodal points on the boundary identified, and the upper unit hemisphere of S 3 with antipodal points on the equator identified. The latter space is clearly diffeomorphic to the unit sphere S 3 with antipodal points identified which coincides with the space of lines in R4 through the origin, that is, with RP3 . The Real Symplectic Group Sp(2n, R). Let 0 I J= . −I 0 Recall that A ∈ L(R2n , R2n ) is symplectic if AT JA gives 1 = det J = (det AT ) · (det AJ) · (det A) = (det A)2 . Hence, det A = ±1, ........................... 15 July 1998—18h02 ...........................

303. 9.2 Some Classical Lie Groups 289 and so, A ∈ GL(2n, R). Furthermore, if A, B ∈ Sp(2n, R), then (AB)T J(AB) = B T AT JAB = J. Hence, AB ∈ Sp(2n, R), and if AT JA = J, then JA = (AT )−1 J = (A−1 )T J, so, T J = A−1 JA−1 or A−1 ∈ Sp(2n, R). Thus, Sp(2n, R) is a group. If a b A= ∈ GL(2n, R), c d then (see Exercise 2.3-2), aT c and bT d are symmetric and A ∈ Sp(2n, R) iﬀ (9.2.12) aT d − cT b = 1. Deﬁne ψ : GL(2n, R) → so(2n) by ψ(A) = AT JA. Let us show that J is a regular value of ψ. Indeed, if A ∈ ψ −1 (J) = Sp(2n, R), the derivative of ψ is Dψ(A) · B = B T JA + AT JB. Now, if C ∈ so(2n), let B = − 1 AJC. 2 We verify, using the identity AT J = JA−1 that Dψ(A) · B = C. Indeed, B T JA + AT JB = B T (A−1 )T J + JA−1 B = (A−1 B)T J + J(A−1 B) = (− 1 JC)T J + J(− 1 JC) 2 2 = − 1 C T JT J − 1 J2 C 2 2 = − 1 CJ2 − 1 J2 C = C 2 2 since JT = −J and J2 = −I. Thus Sp(2n, R) = ψ −1 (J) is a closed smooth submanifold of GL(2n, R) whose Lie algebra is ker Dψ(I) = {B ∈ L R2n , R2n | B T J + JB = 0}. Sp(2n, R) is called the symplectic group and its Lie algebra sp(2n, R) = {A ∈ L R2n , R2n | AT J + JA = 0} ........................... 15 July 1998—18h02 ...........................

304. 290 9. An Introduction to Lie Groups the symplectic algebra. Moreover, if a b A= ∈ sl(2n, R), c d then A ∈ sp(2n, R) iff d = −aT , c = cT , and b = bT . (9.2.13) The dimension of sp(2n, R) can be readily calculated to be 2n + n. 2 Using (9.2.12) it follows that all matrices of the form I 0 tI I √ are symplectic. However, the norm of such a matrix is equal to 2n + nt2 , which is unbounded if t ∈ R. Therefore, Sp(2n, R) is not a bounded subset of gl(2n, R) and hence, is not compact. Proposition 9.2.11. Sp(2n, R) := {A ∈ GL(2n, R) | AT JA = J} is a noncompact, connected Lie group of dimension 2n2 + n. Its Lie algebra sp(2n, R) consists of the 2n×2n matrices A satisfying AT J+JA = 0, where 0 I J= −I 0 with I the n × n identity matrix. sentence We shall indicate in §9.3 how one proves that Sp(2n, R) is connected. starts with Recall that the symplectic group is related to classical mechanics as follows. symbol In order to gain a better understanding of Sp(n, R) we shall address below their eigenvalues. Lemma 9.2.12. If A ∈ Sp(n, R), then det A = 1. Proof. Since A JA = J and det J = 1 it follows that (det A)2 = 1. T Unfortunately, this still leaves open the possibility that det A = −1. To eliminate it, we proceed in the following way. Define the symplectic form Ω on R2n by Ω(u, v) = uT Jv, that is, relative to the chosen basis of R2n , the matrix of Ω is J. Define a volume form µ on R2n by µ(v1 , . . . v2n ) = det (Ω(vi , vj )) . By the definition of the determinant of a linear map, (det A)µ = A∗ µ, we get (det A)µ (v1 , . . . , v2n ) = (A∗ µ) (v1 , . . . , v2n ) = µ (Av1 , . . . , Av2n ) = det (Ω (Avi , Avj )) = det (Ω (vi , vj )) = µ (v1 , . . . , v2n ) ........................... 15 July 1998—18h02 ...........................

305. 9.2 Some Classical Lie Groups 291 since A ∈ Sp(2n, R), which is equivalent to Ω(Au, Av) = Ω(u, v) for all u, v ∈ R2n . Taking for v1 , . . . , v2n the standard basis of R2n we conclude that det A = 1. Proposition 9.2.13 (Symplectic Eigenvalue Theorem). If λ0 ∈ C is an eigenvalue of A ∈ Sp(2n, R) of multiplicity k, then 1/λ0 , λ0 , and 1/λ0 are eigenvalues of A of the same multiplicity k. Moreover, if ±1 occur as eigenvalues, their multiplicities are even. Proof. Since A is a real matrix, if λ0 is an eigenvalue of A of multiplicity k, so is λ0 by elementary algebra. Let us show that 1/λ0 is also an eigenvalue of A. If p(λ) = det(A − λI) is the characteristic polynomial of A, since T JAJ−1 = A−1 , det J = 1, J−1 = −J = JT , and det A = 1 ( by Lemma 9.2.11), we get p(λ) = det(A − λI) = det J(A − λI)J−1 T = det(JAJ−1 − λI) = det A−1 − λI = det(A−1 − λI) = det A−1 (I − λA) = det(I − λA) = det(λ( λ I − A)) 1 = λ2n det( λ I − A) 1 = λ2n (−1)2n det(A − λ I) 1 = λ2n p(λ). (9.2.14) Since 0 is not an eigenvalue of A, it follows that 1 p(λ) = 0 iﬀ p λ = 0, and hence, λ0 is an eigenvalue of A iﬀ 1/λ0 is an eigenvalue of A. Now assume that λ0 has multiplicity k, that is, p(λ) = (λ − λ0 )k q(λ) for some polynomial q(λ) of degree 2n − k satisfying q(λ0 ) = 0. Since p(λ) = λ2n p(1/λ), we conclude that k p(λ) = p 1 λ λ2n = (λ − λ0 )k q(λ) = (λλ0 )k 1 λ0 − 1 λ q(λ). However, λk 0 q(λ) λ2n−k ........................... 15 July 1998—18h02 ...........................

306. 292 9. An Introduction to Lie Groups is a polynomial in 1/λ, since the degree of q(λ) is 2n − k, k ≤ 2n. Thus 1/λ0 is a root of p(λ) having multiplicity l ≥ k. Reversing the roles of λ0 and 1/λ0 , we similarly conclude that k ≥ l and hence, it follows that k = l. Finally, note that λ0 = 1/λ0 iﬀ λ0 = ±1. Thus, since all eigenvalues of A occur in pairs whose product is 1 and the size of A is (2n) × (2n), it follows that the total number of times +1 and −1 occur as eigenvalues an even number of times. However, since det A = 1 by Lemma 9.2.12, we conclude that −1 occurs an even number of times as an eigenvalue of A ( if it occurs at all). Therefore, the multiplicity of 1 as an eigenvlaue of A, if it occurs, is also even. Figure 9.2.1 illustrates all possible conﬁgurations of the eigenvalues of A ∈ Sp(4, R). y y x x complex saddle saddle center y y x x real saddle generic center y y y (2) x x x (2) (2) (4) (2) degenerate saddle identity degenerate center Figure 9.2.1. Symplectic Eigenvalue Theorem on R4 . Next, we study the eigenvalues of the matrices in sp(2n, R). The following theorem is useful in the stability analysis of relative equilibria. If A ∈ sp(2n, R), then AT J + JA = 0 so that if p(λ) = det(A − λI) is the ........................... 15 July 1998—18h02 ...........................

307. 9.2 Some Classical Lie Groups 293 characteristic polynomial of A, we have p(λ) = det(A − λI) = det(J(A − λI)J) = det(JAJ − λI) = det(−AT J2 + λI) = det(AT + λI) = det(A + λI) = p(−λ). In particular, notice that trace(A) = 0. Proceeding as before and using this identity, we conclude the following: Title too long. Proposition 9.2.14 (Infinitesimally Symplectic Eigenvalue Theorem). If λ0 ∈ C is an eigenvalue of A ∈ sp(2n, R) of multiplicity k, then −λ0 , λ0 , and −λ0 are eigenvalues of A of the same multiplicity k. Moreover, if 0 is an eigenvalue, it has even multiplicity. Figure 9.2.2 shows the possible infinitesimally symplectic eigenvalue configurations for A ∈ sp(4, R). y y x x complex saddle saddle center y y x x real saddle generic center y y y (2) x x x (2) (2) (4) (2) degenerate saddle identity degenerate center Figure 9.2.2. Infinitesimally symplectic Eigenvalue Theorem on R4 . ........................... 15 July 1998—18h02 ...........................

308. 294 9. An Introduction to Lie Groups The Symplectic Group and Mechanics. Consider a particle of mass m moving in a potential V (q), where q = (q 1 , q 2 , q 3 ) ∈ R3 . Newton’s second law states that the particle moves along a curve q(t) in R3 in such a way that m¨ = − grad V (q). Introduce the momentum pi = mq i , i = 1, 2, 3, q ˙ and the energy 3 1 H(q, p) = p2 + V (q). 2m i=1 i Then ∂H ∂V ∂H 1 i = i = −m¨ i = −pi , and q ˙ = pi = q i , ˙ ∂q ∂q ∂pi m and hence Newton’s law F = ma is equivalent to Hamilton’s equations ∂H ∂H qi = ˙ , pi = − ˙ , i = 1, 2, 3. ∂pi ∂q i Writing z = (q, p),   ∂H 0 I  ∂q  J · grad H(z) =   ˙ ˙ −I 0  ∂H  = (q, p) = z, ˙ ∂p so Hamilton’s equations read z = J · grad H(z). Now let ˙ f : R 3 × R3 → R 3 × R 3 and write w = f (z). If z(t) satisfies Hamilton’s equations z = J · grad H(z), ˙ then w(t) = f (z(t)) satisfies w = AT z, where AT = [∂wi /∂z j ] is the ˙ ˙ Jacobian matrix of f . By the chain rule, w = AT J gradz H(z) = AT JA gradw H(z(w)). ˙ Thus, the equations for w(t) have the form of Hamilton’s equations with energy K(w) = H(z(w)) if and only if AT JA = J; that is, iff A is symplectic. A nonlinear transformation f is canonical iff its Jacobian is symplectic. As a special case, consider a linear map A ∈ Sp(2n, R) and let w = Az. Suppose H is quadratic, that is, of the form H(z) = z, Bz /2, where B is a symmetric (2n × 2n) matrix. Then grad H(z) · δz = 1 δz, Bz + z, Bδz 2 = 1 ( δz, Bz + Bz, δz ) = δz, Bz , 2 so grad H(z) = Bz and thus the equations of motion become the linear equations z = JBz. Now ˙ w = Az = AJBz = J(AT )−1 Bz = J(AT )−1 BA−1 Az = JB w, ˙ ˙ ........................... 15 July 1998—18h02 ...........................

309. 9.2 Some Classical Lie Groups 295 where B = (AT )−1 BA−1 is symmetric. For the new Hamiltonian we get H (w) = w, (AT )−1 BA−1 w = A−1 w, BA−1 w = H(A−1 w) = H(z). Thus, Sp(2n, R) is the linear invariance group of classical mechanics. The Complex General Linear Group GL(n, C). Many important Lie groups involve complex matrices. As in the real case, GL(n, C) = {n × n invertible complex matrices} is an open set in L(Cn , Cn ) = {n × n complex matrices}. Clearly GL(n, C) is a group under matrix multiplication. Therefore, GL(n, C) is a Lie group, and has a Lie algebra gl(n, C) = {n × n complex matrices} = L(Cn , Cn ). Hence GL(n, C) has complex dimension n2 , that is, real dimension 2n2 . We shall prove below that GL(n, C) is connected (contrast this with the fact that GL(n, R) has two compnents). As in the real case, we will need a polar decomposition theorem to do this. A matrix U ∈ GL(n, C) is T unitary if U U † = U † U = I, where U † := U . A matrix P ∈ gl(n, C) is Hermitian, if P † = P . A Hermitian matrix P is called positive definite, denoted P 0, if P z, z 0 for all z ∈ Cn , z = 0, where , denotes the inner product on Cn . Note that P 0 implies that P is invertible. Proposition 9.2.15 (Complex Polar Decomposition Theorem). For any A ∈ GL(n, C), there exists a unique unitary matrix U and positive definite matrices P1 , P2 such that A = U P1 = P2 U, where P1 = U † P2 U . The proof is identical to that of Proposition 9.2.1 with the obvious changes. The only additional property needed is the fact that the eigenvalues of a Hermitian matrix are real . As in the proof of the real case, one needs to use the connectedness of the space of unitary matrices, to be proved later. Proposition 9.2.16. The group GL(n, C) is a complex noncompact connected Lie group of complex dimension n2 and real dimension 2n2 . Its Lie algebra gl(n, C) consists of all n × n complex matrices with the commutator bracket. On gl(n, C), the inner product is defined by A, B = trace(AB † ). ........................... 15 July 1998—18h02 ...........................

310. 296 9. An Introduction to Lie Groups The Complex Special Linear Group SL(n, C) := {A ∈ GL(n, C) | det A = 1} is treated as in the real case. In the proof of its connectedness one uses the Complex Polar Decomposition Theorem and the fact that any Hermitian matrix can be diagonalized by conjugating it with an appropriate unitary matrix. Proposition 9.2.17. The group SL(n, C) is a complex noncompact Lie group of complex dimension n2 − 1 and real dimension 2(n2 − 1). Its Lie algebra sl(n, C) consists of all n × n complex matrices of trace zero with the commutator bracket. The Unitary Group U (n). Recall that Cn has the Hermitian inner product: n x, y = xi y i , ¯ i=0 where x = x , . . . , x ∈ C , and y = y 1 , . . . , y n ∈ Cn , and y i denotes 1 n n ¯ the complex conjugate. Let U(n) = {A ∈ GL(n, C) | Ax, Ay = x, y }. The orthogonality condition Ax, Ay = x, y is equivalent to AA† = A† A = I, where A† = AT , that is, Ax, y = x, A† y . From | det A| = 1, ¯ we see that det maps U(n) into the unit circle S 1 = {z ∈ C | |z| = 1}. As is to be expected by now, U(n) is a closed Lie subgroup of GL(n, C) with Lie algebra u(n) = {A ∈ L(Cn , Cn )| Ax, y = − x, Ay } = {A ∈ gl(n, C) | A† = −A}; the proof parallels that for O(n). The elements of u(n) are called skew- Hermitian matrices. Since the norm of A ∈ U (n) is 1/2 √ A = trace(A† A) = (trace I)1/2 = n, it follows that U (n) is closed and bounded, hence compact, in GL(n, C). From the deﬁnition of u(n) it immediately follows that the real dimension of U (n) is n2 . Thus, even though the entries of the elements of U (n) are complex, U (n) is a real Lie group. In the special case n = 1, a complex linear map ϕ : C → C is multiplication by some complex number z, and ϕ is an isometry if and only if |z| = 1. In this way the group U(1) is identiﬁed with the unit circle S 1 . The special unitary group SU(n) = {A ∈ U(n) | det A = 1} = U(n) ∩ SL(n, C) ........................... 15 July 1998—18h02 ...........................

311. 9.2 Some Classical Lie Groups 297 is a closed Lie subgroup of U(n) with Lie algebra su(n) = {A ∈ L(Cn , Cn ) | Ax, y = − x, Ay and tr A = 0}. Hence, SU(n) is compact and has (real) dimension n2 − 1. We shall prove later that both U(n) and SU(n) are connected. Proposition 9.2.18. The group U(n) is a compact real Lie subgroup of GL(n, C) of (real) dimension n2 . Its Lie algebra u(n) consists of the space of skew-Hermitian n × n matrices with the commutator bracket. SU(n) is a closed real Lie subgroup of U(n) of dimension n2 − 1 whose Lie algebra su(n) consists of all trace zero skew-Hermitian n × n matrices. We now want to relate Sp(2n, R), O(2n), and U(n). To do this, we identify Cn = Rn ⊕ iRn and we express the Hermitian inner product on Cn as a pair of real bilinear forms, namely, if x1 + iy1 , x2 + iy2 ∈ Cn , for x1 , x2 , y1 , y2 ∈ Rn , then x1 + iy1 , x2 + iy2 = x1 , y1 + x2 , y2 + i ( x2 , y1 − x1 , y2 ) . Thus, identifying Cn with Rn × Rn and C with R × R, we can write I 0 y1 0 I y1 (x1 , y1 ), (x2 , y2 ) = (x1 , x2 ) , −(x1 , x2 ) . 0 I y2 −I 0 y2 (9.2.15) The next task is to represent elements of U(n) as (2n) × (2n) matrices with real entries. Since U(n) is a closed subgroup of GL(n, C) we begin by representing the elements of gl(n, C) in this way. Let A + iB ∈ gl(n, C) with A, B ∈ gl(n, R) and let x + iy ∈ CM . Then (A + iB)(x + iy) = (Ax − By) + i(Ay + Bx) suggest that A −B A + iB ∈ GL(n, C) → ∈ GL(2n, R) (9.2.16) B A is the desired embedding of GL(n, C) into GL(2n, R) . It is indeed straightforward to verify that the above map is an injective Lie group homomorphism, so we can identify GL(n, C) with all invertible (2n) × (2n) matrices of the form A −B (9.2.17) B A with A, B ∈ gl(n, R). Therefore, U(n) is embedded in GL(n, R) as the set of matrices of the form (9.2.17) with a certain additional property to be ........................... 15 July 1998—18h02 ...........................

312. 298 9. An Introduction to Lie Groups determined below. If A + iB ∈ U(n) then (A + iB)† (A + iB) = I. However, under the homomorphism (9.2.16) (A + iB)† = AT − iB T is sent to the matrix AT BT . −B T AT Therefore, (A + iB)† (A + iB) = I becomes I 0 AT BT A −B = 0 I −B T AT B A AT A + B T B −AT B + B T A = −B T A + AT B B T B + AT A which is equivalent to AT A + B T B = I and AT B is symmetric. (9.2.18) Proposition 9.2.19. Sp(2n, R) ∩ O(2n, R) = U(n). Proof. We have seen that A + iB ∈ U(n) iﬀ (9.2.18) holds. Now let us characterize all matrices of the form A B ∈ Sp(2n, R) ∩ O(2n, R). C D By (9.2.12) we need to have AT D − C T B = I and AT C, B T D symmetric. (9.2.19) Since this matrix is also in O(2n), we have I O A B AT CT = O I C D BT DT AAT + BB T AC T + BDT = CAT + DB T CC T + DDT which is equivalent to AAT + BB T = I, AC T + BDT = 0, CC T + DDT = I. (9.2.20) ........................... 15 July 1998—18h02 ...........................

313. 9.2 Some Classical Lie Groups 299 Now, multiply on the right by D the first identity in (9.2.20), to get from (9.2.19) D = AAT D + BB T D = A(I + C T B) + BB T D = A + AC T B + BDT B = A + (AC T + BDT )B = A by the second identity in (9.2.20). Next, multiply on the right by B the last identity in (9.2.20) and use, as before (9.2.19) to get B = CC T B + DDT B = C(AT D − I) + DDT B = CAT D − C + DB T D = −C + (CAT + DB T )D = −C by the second identity in (9.2.20). We have thus shown that A B ∈ Sp(2n, R) ∩ O(2n) C D iff A = D, B = −C, AT A + C T C = I, and AT C is symmetric, which coincide with the conditions (9.2.18) characterizing U(n). The Group SU(2) warrants special attention since it appears in many physical applications such as the Cayley–Klein parameters for the free rigid body or in the construction of the (non-Abelian) gauge group for the Yang– Mills equations in elementary particle physics. From the general formula for the dimension of SU(n) it follows that dim SU(2) = 3. Also, SU(2) is diffeomorphic to the three-sphere S 3 = {x ∈ R4 | x = 1}, with the diffeomorphism given by x0 − ix3 −x2 + ix1 x = (x0 , x1 , x2 , x3 ) ∈ S 3 ⊂ R4 → ∈ SU(2). x2 + ix1 x0 − ix3 (9.2.21) Therefore, SU(2) is connected and simply connected. By Euler’s Theorem [?] every element of SO(3) different from the identity is determined by a vector v, which we can choose to be a unit vector, and an angle of rotation θ about that axis. The trouble is, the pair (v, θ) and (−v, −θ) represent the same rotation and there is no consistent way to continuously choose one of these pairs, valid for the entire group SO(3). Such a choice is called, in physics, a choice of spin. This immediately suggests the existence of a double cover of SO(3), that, hopefully, should also ........................... 15 July 1998—18h02 ...........................

314. 300 9. An Introduction to Lie Groups be a Lie group. We will show below that SU(2) fulfills these requirements. This is based on the following construction. Let σ1 , σ2 , σ3 be the Pauli spin matrices, defined by 0 1 0 −i 1 0 σ1 = , σ2 = , and σ3 = , 1 0 i 0 0 −1 and let σ = (σ1 , σ2 , σ3 ). Then one checks that [σ1 , σ2 ] = 2iσ3 (plus cyclic permutations) from which one finds that the map 1 1 −ix3 −ix1 − x2 x→x= ˜ x·σ = , 2i 2 −ix1 + x2 ix3 where x · σ = x1 σ1 + x2 σ2 + x3 σ3 , is a Lie algebra isomorphism between R3 and the (2 × 2) skew-Hermitian traceless matrices (the Lie algebra of SU(2)); that is, [˜ , y] = (x × y)˜. Note that x ˜ − det(x · σ) = x 2 , and trace (˜ y) = − 1 x · y. x˜ 2 Define the Lie group homomorphism π : SU(2) → GL(3, R) by (π(A)x) · σ = A(x · σ)A† = A(x · σ)A−1 . (9.2.22) A straightforward computation, using the expression (9.2.21) shows that ker π = {±I}. Therefore, π(A) = π(B) if and only if A = ±B. Since π(A)x 2 = − det((π(A)x) · σ) = − det(A(x · σ)A−1 ) = − det(x · σ) = x 2 , it follows that π(SU(2)) ⊂ O(3). But π(SU(2)) is connected, being the continuous image of a connected space, and so π(SU(2)) ⊂ SO(3). Let us show that π : SU(2) → SO(3) is a local diffeomorphism. Indeed, if α ∈ su(2), then ˜ (Te π(α)x) · σ = (x · σ)α† + α(x · σ) ˜ ˜ ˜ = [α, x · σ] = 2i[α, x] ˜ ˜ ˜ = 2i(α × x)˜ = (α × x) · σ, ˜ ˜ = (αx) · σ. ˆ ........................... 15 July 1998—18h02 ...........................

315. 9.2 Some Classical Lie Groups 301 ˜ ˆ that is, Te π(α) = α. Thus, Te π : su(2) −→ so(3) is a Lie algebra isomorphism and hence is a local diffemorphism in a neighborhood of the identity. Since π is a Lie group homomorphism it is a local diffeomorphism around every point. In particular, π(SU(2)) is open and hence closed (its complement is a union of open cosets) in SO(3)). Since it is nonempty and SO(3)) is connected, we have π(SU(2)) = SO(3). Therefore, π : SU(2) → SO(3) is a 2 to 1 surjective submersion. Summarizing, we have the commutative diagram in Figure 9.2.1. ≈ S3 E SU(2) 2:1 2:1 c c RP3 E SO(3) ≈ Figure 9.2.3. The link between SU(2) and SO(3). Proposition 9.2.20. The Lie group SU(2) is the simply connected 2 to 1 covering group of SO(3). Quaternions. The division ring H (or, by abuse of language, the non- commutative field) of quaternions is generated over the reals by three elements i, j, k with the relations i2 = j2 = k2 = −1 ij = −ji = k, jk = −kj = i, ki = −ik = j. Quaternionic multiplication is performed in the usual manner (like polynomial multiplication) taking the above relations into account. If a ∈ H, we write a = (as , av ) = as + a1 i + a2 j + a3 k v v v for the scalar and vectorial part of the quaternion, where as , a1 , a2 , v v a3 ∈ R. Quaternions having zero scalar part are also called pure quater- v nions. With this notation, quaternionic multiplication has the expression ab = (as bs − av · bv , as bv + bs av + av × bv ) ........................... 15 July 1998—18h02 ...........................

316. 302 9. An Introduction to Lie Groups In addition, every quaternion a = (as , av ) has a conjugate a := (as , −av ), that is, the real numbers are ﬁxed by the conjugation and i = −i, j = −j, and k = −k. Every quaternion a = 0 has an inverse given by a−1 = a/|a|2 , where |a|2 := aa = aa = a2 + av 2 . s In particular, the unit quaternions, which, as a set, equal the unit sphere S 3 in R4 , form a group under quaternionic multiplication. Proposition 9.2.21. The unit quaternions S 3 = {a ∈ H | |a| = 1} form a Lie group isomorphic to SU(2) via the isomorphism (9.2.21). Proof. We already noted that (9.2.21) is a diﬀeomorphism of S 3 with SU(2), so all that remains to be shown is that it is a group homomorphism which is a straightforward computation. Since the Lie algebra of S 3 is the tangent space at 1, it follows that it is isomorphic to the pure quaternions R3 . We begin by determining the adjoint action of S 3 on its Lie algebra. If a ∈ S 3 and bv is a pure quaternion, the derivative of the conjugation is given by a Ada bv = abv a−1 = abv |a|2 1 = (−av · bv , as bv + av × bv )(as , −av ) |a|2 1 = 2 0, 2as (av × bv ) + 2(av · bv )av + (a2 − av 2 )bv . |a| s Therefore, if a(t) = (1, tav ), we have a(0) = 1, a (0) = av , so that the Lie bracket on the pure quaternions R3 is given by d [av , bv ] = Ada(t) bv dt t=o d 1 = (2t(av × bv ) + 2t2 (av · bv )av + 1 − t2 av 2 )bv dt t=0 1 + t2 av 2 = 2av × bv . Thus, the Lie algebra of S 3 is R3 relative to the Lie bracket given by twice the cross product of vectors. The derivative of the Lie group isomorphism (9.2.21) is given by −ix3 −ix1 − x2 x ∈ R3 → = 2˜ ∈ su(2), x −ix1 + x2 ix3 and is thus a Lie algebra isomorphism from R3 with twice the cross product as bracket to su(2), or equivalently to (R3 , x). ........................... 15 July 1998—18h02 ...........................

317. 9.2 Some Classical Lie Groups 303 Let us return to the commutative diagram in Figure 9.2.1 and determine explicitly the 2 to 1 surjective map S 3 → SO(3) that associates to a quaternion a ∈ S 3 ⊂ H the rotation matrix A ∈ SO(3). To compute this map, let a ∈ S 3 and associate to it the matrix as − ia3 v −a2 − ia1 v v U= , a2 − ia1 v v as + ia3 v where a = (as , av ) = (as , a1 , a2 , a3 ). By (9.2.22), the rotation matrix is v v v given by A = π(U ), namely, (Ax) · σ = (π(U )x) · σ = U (x · σ)U † as − ia3 v −a2 − ia1 v v x3 x1 − ix2 = a2 − ia1 v v as + ia3 v x + ix2 1 −x3 as + ia3 v a2 + ia1 v v −a2 + ia1 v v as − ia3 v = a2 + (a1 )2 − (a2 )2 − (a3 )2 x1 + 2(a1 a2 − as a3 )x2 s v v v v v v +2(as a2 + a1 a3 )x3 σ1 v v v + 2 a1 a2 + as a3 x1 + a2 − (a1 )2 + (a2 )2 − (a3 )2 x2 v v v s v v v +2 a2 a3 − as a1 x3 σ2 v v v + 2 a1 a3 − as a2 x1 + as a1 + a2 a3 x2 v v v v v v + a2 − (a1 )2 − (a2 )2 + (a3 )2 x3 σ3 . s v v v Thus, taking into account that a2 + (a1 )2 + (a2 )2 + (a3 )2 = 1, we get the s v v v expression of the matrix A as   2a2 + 2(a1 )2 − 1 s v 2(−as a3 + a1 a2 ) v v v 2(as a2 + a1 a3 ) v v v  2(as a3 + a1 a2 ) v v v 2a2 + 2(a2 )2 − 1 s v 2(−as a1 + a2 a3 ) v v v 2(−as a1 + a2 a3 ) v v v 2(as a1 + a2 a3 ) v v v 2a2 + (a3 )2 − 1 s v = (2a2 − 1)I + 2as av + 2av ⊗ av , s ˆ (9.2.23) where av ⊗ av is the symmetric matrix whose (i, j) entry equals ai aj . The v v map a ∈ S 3 → (2a2 − 1)I + 2as av + 2av ⊗ av s ˆ is called the Euler–Rodrigues parametrization. It has the advantage, as opposed to the Euler angles parametrization, which has a coordinate singularity, of being global. This is of crucial importance in computational mechanics (see Marden and Wendlandt [1997]). Finally, let us rewrite Rodrigues’ formula (9.2.8) in terms of unit quaternions. Let ω ω a = (as , av ) = cos , sin n , 2 2 ........................... 15 July 1998—18h02 ...........................

318. 304 9. An Introduction to Lie Groups where ω 0 is an angle and n is a unit vector. Since n2 = n ⊗ n − I, from ˆ (9.2.8) we get exp(ωn) ω = I + (sin ω)ˆ + 2 sin2 n (n ⊗ n − I) 2 ω ω ω ω = 1 − 2 sin2 I + 2 cos sin n + 2 sin2 ˆ n⊗n 2 2 2 2 = 2a2 − 1 I + 2as av + 2av ⊗ av . s ˆ This expression then produces a rotation associated to each unit quaternion a. In addition, using this parametrization, Rodrigues [1840] found a beautiful way of expressing the product of two rotations exp(ω1 η 1 ) · exp(ω2 η 2 ) in terms of the given data. In fact, this was an early exploration of the spin group! We refer to Whittaker [1927], §7, Altmann [1986], Enos [1993], Simo and Lewis [1994] and references therein for further information. SU(2) Conjugacy Classes and the Hopf Fibration. We next determine all conjugacy classes of S 3 SU(2). If a ∈ S 3 , then a−1 = a and a straightforward computation gives aba−1 = (bs , 2(av · bv )av + 2as (av × bv ) + (2a2 − 1)bv ) s for any b ∈ S 3 . If bs = ±1, that is, bv = 0, then the above formula shows that aba−1 = b for all a ∈ S 3 , that is, the classes of I and −I, where I = (1, 0), each consist of one element and the center of SU(2) S 3 is {±I}. In what follows, assume that bs = ±1, or, equivalently, that bv = 0 and fix this b ∈ S 3 throughout the following discussion. We shall prove that, given x ∈ R3 with x = |bv , we can find a ∈ S 3 such that 2(av · bv )av + 2as (av × bv ) + (2a2 − 1)bv = x. s (9.2.24) If x = cbv for some c = 0, then the choice av = 0 and 2a2 = 1 + c satisfies s (9.2.24). Now assume that x and bv are not collinear. Take the dot product of (9.2.24) with bv and get: 2(av · bv )2 + 2a2 bv s 2 = bv 2 + x · bv . If bv 2 + x · bv = 0, since bv = 0, it follows that av · bv = 0 and as = 0. Returning to (9.2.24) it follows that −bv = x, which is excluded. Therefore, x·bv + bv 2 = 0 and searching for av ∈ R3 such that av ·bv = 0, it follows that x · bv + bv 2 a2 = s = 0. 2 bv 2 Now, take the cross product of (9.2.24) with bv and recall that we assumed av · bv = 0 to get 2as bv 2 av = bv × x, ........................... 15 July 1998—18h02 ...........................

319. 9.2 Some Classical Lie Groups 305 whence bv × x av = , 2as bv 2 which is allowed, since bv = 0 and as = 0. Note that a = (as , av ) just determined satisfies av · bv = 0 and |a|2 = a2 + av s 2 =1 since x = bv . Proposition 9.2.22. The conjugacy classes of S 3 SU(2) are the two- spheres {bv ∈ R3 | bv 2 = 1 − b2 } s for each bs ∈ [−1, 1], which degenerate to the North and South poles (±1, 0, 0, 0) comprising the center of SU(2). The above proof shows that any unit quaternion is conjugate in S 3 to a quaternion of the form as + a3 k, as , a3 ∈ R, which in terms of matrices v v and the isomorphis (9.2.21) says that any SU(2) matrix is conjugate to a diagonal matrix . The conjugacy class of k is the unit sphere in S 2 and the orbit map π : S3 → S2, π(a) = aka is the Hopf fibration. The subgroup H = {as + a3 k ∈ S 3 | as , a3 ∈ R} ⊂ S 3 v v is a closed, one-dimensional Abelian Lie subgroup of S 3 isomorphic via (9.2.21) to the set of diagonal matrices in SU(2) and is hence a circle S 1 . Note that the isotropy of k in S 3 consists of H, as an easy computation, using (9.2.24) shows. Therefore, since the orbit of k is diffeomorphic to S 3 /H it follows that the fibers of the Hopf fibration equal the left cosets aH for a ∈ S 3 . Finally, we shall give an expression of the Hopf fibration in terms of complex variables. In the representation (9.2.21), set w1 = x2 + ix1 , w2 = x0 + ix3 , and note that if a = (x0 , x1 , x2 , x3 ) ∈ S 3 ⊂ H, then aka corresponds to x0 − ix3 −x2 − ix1 −i 0 x0 + ix3 x2 + ix1 x2 − ix1 x0 + ix3 0 i −x2 + ix1 x0 − ix3 −i |x0 + ix3 |2 − |x2 + ix1 |2 −2i x2 + ix1 (x0 − ix3 ) = −2i(x2 − ix1 )(x0 + ix3 ) i |x0 + ix3 |2 − |x2 + ix1 |2 ........................... 15 July 1998—18h02 ...........................

320. 306 9. An Introduction to Lie Groups Thus, if we consider the diffeomorphisms x0 − ix3 −x2 − ix1 (x0 , x1 , x2 , x3 ) ∈ S 3 ⊂ H → ∈ SU(2) x2 − ix1 x0 + ix3 → −i(x2 + ix1 ), −i(x0 + ix3 ) ∈ S 3 ⊂ C2 the above orbit map, that is, the Hopf fibration, becomes (w1 w2 ) ∈ S 3 → 2w1 w2 , |w2 |2 − |w1 |2 ∈ S 2 . The Unitary Symplectic Group Sp(2n). In complete analogy to Rn and Cn we define Hn = {a = (a1 , . . . , an ) | ai ∈ H}. This satisfies all axioms of an n-dimensional vector space of H with the sole exception that H is not a field, being non-commutative. We want to construct a group analogous to O(n) when we worked with Rn , or to U(n) when we worked with Cn . For this, we introduce the quaternionic inner product n a, b H = ap bp , p=1 where a, b ∈ Hn and bp is the quaternion conjugate to bp , for p = 1, . . . , n. Again, the usual axioms for the inner product are satisfied, by being careful in the scalar multiplication by quaternions, that is, (i) a1 + a2 , b = a1 , b + a2 , b , (ii) αa, b = α a, b and a, bα = a, b α, for all α ∈ H, (iii) a, b = b, a , (iv) a, a ≥ 0 and a, a = 0 iff a = 0. The next step is to introduce the analogue of the usual matrix multiplication and to insure its linearity. Again, because of non-commutativity of H, care has to be taken with this. Define the anaologue of a linear map given by a matrix by T : Hn → Hn , n (T a)r = tpr ar , p=1 for a given n × n matrix [tpr ]. It is straightforward to note that T (aα) = (T a)α, for any α ∈ H, but that T (αa) = α(T a), in general. Therefore, usual matrix multiplication is a right-linear map and, in general, it is not left-linear over H. ........................... 15 July 1998—18h02 ...........................

321. 9.2 Some Classical Lie Groups 307 As real vector spaces, C2n and Hn are isomorphic. However, there is a lot of structure that we shall exploit below by realizing left quaternionic matrix multiplication as a complex linear map. To achieve this, we shall identify, as before, i ∈ C with the quaternion i ∈ H and will define the fundamental right complex isomorphism X : C2n → Hn by X(u, v) = u + jv, where u, v ∈ Cn , and we regard C embedded in H by x + iy → x + iy, for x, y ∈ R. We have X((u, v)α) = X(u, v)α for all α ∈ C. So, again, we get only right linearity. The key property of X is that it turns a left quaternionic matrix multiplication operator into a usual complex linear operator on C2n . Indeed, if [tpr ] is a quaternionic n × n matrix, then X −1 T X : C2n → C2n is complex linear . To verify this, let α ∈ C, u, v ∈ Cn and note that X −1 T X (α(u, v)) = X −1 T X ((u, v)α) = X −1 T ((X(u, v))α) = X −1 ((T X(u, v))α) = (X −1 T X(u, v))α = α(X −1 T X(u, v)). Let us determine, for example, the 2n × 2n complex matrix J that corresponds to the right linear quaternionic map given by the diagonal map jI. We have J(u, v) = (X −1 jIX)(u, v) = (X −1 jI)(u + jv) = X −1 (ju, −v) = (−v, u), that is, 0 I J= −I 0 is the canonical symplectic structure on Cn ×Cn = C2n . Define the injective map between the space of right linear quaternionic maps on Hn defined by left multiplication by a matrix to the space of complex linear maps on C2n by T → TX := X −1 T X. Among all the complex linear maps C2n → C2n we want to characterize those that correspond to left matrix multiplication on Hn . ........................... 15 July 1998—18h02 ...........................

322. 308 9. An Introduction to Lie Groups Exercises 9.2-1. Describe the set of matrices in SO(3) that are also symmetric. 9.2-2. If A ∈ Sp(2n, R), show that AT ∈ Sp(2n, R) as well. Check soln. to 9.2-3. Show that Sp(2n, R) ∩ SO(2n) = SU(n). 9.2-3 in view of new text. 9.2-4. Show that sp(2n, R) is isomorphic, as a Lie algebra, to the space of homogeneous quadratic functions on R2n under the Poisson bracket. 9.2-5. A map f : Rn → Rn preserving the distance between any two points, that is, f (x) − f (y) = x − y for all x, y ∈ Rn , is called an isometry. Show that f is an isometry preserving the origin if and only if f ∈ O(n). 9.3 Actions of Lie Groups In this section we develop some basic facts about actions of Lie groups on manifolds. One of our main applications later will be the description of Hamiltonian systems with symmetry groups. Basic Definitions. We begin with the definition of the action of a Lie group G on a manifold M . Definition 9.3.1. Let M be a manifold and let G be a Lie group. A (left) action of a Lie group G on M is a smooth mapping Φ : G × M → M such that: (i) Φ(e, x) = x, for all x ∈ M ; and (ii) Φ(g, Φ(h, x)) = Φ(gh, x), for all g, h ∈ G and x ∈ M . A right action is a map Ψ : M × G → M that satisfies Ψ(x, e) = x and Ψ(Ψ(x, g), h) = Ψ(x, gh). We sometimes use the notation g · x = Φ(g, x) for left actions, and x · g = Ψ(x, g) for right actions. In the infinite-dimensional case there are important situations where care with the smoothness is needed. For the formal development we assume we are in the Banach-Lie group context. For every g ∈ G let Φg : M → M be given by x → Φ(g, x). Then (i) becomes Φe = idM while (ii) becomes Φgh = Φg ◦ Φh . Definition 9.3.1 can now be rephrased by saying that the map g → Φg is a homomorphism of G into Diff(M ), the group of diffeomorphisms of M . In the special but important case where M is a Banach space V and each Φg : V → V is a continuous linear transformation, the action Φ of G on V is called a representation of G on V . ........................... 15 July 1998—18h02 ...........................

323. 9.3 Actions of Lie Groups 309 Examples (a) SO(3) acts on R3 by (A, x) → Ax. This action leaves the two-sphere S 2 invariant, so the same formula defines an action of SO(3) on S 2 . (b) GL(n, R) acts on Rn by (A, x) → Ax. (c) Let X be a complete vector field on M , that is, one for which the flow Ft of X is defined for all t ∈ R. Then Ft : M → M defines an action of R on M . Orbits and Isotropy. If Φ is an action of G on M and x ∈ M , the orbit of x is defined by Orb(x) = {Φg (x) | g ∈ G} ⊂ M. In finite dimensions one can show that Orb(x) is an immersed submanifold of M (Abraham and Marsden [1978, p. 265]). For x ∈ M , the isotropy (or stabilizer or symmetry ) group of Φ at x is given by Gx := {g ∈ G | Φg (x) = x} ⊂ G. Since the map Φx : G → M defined by Φx (g) = Φ(g, x) is continuous, Gx = (Φx )−1 (x) is a closed subgroup and hence a Lie subgroup of G. The manifold structure of Orb(x) is defined by requiring the bijective map [g] ∈ G/Gx → g · x ∈ Orb(x) to be a diffeomorphism. That G/Gx is a smooth manifold follows from Proposition 9.3.2, which is discussed below. An action is said to be: 1. transitive if there is only one orbit or, equivalently, if for every x, y ∈ M there is a g ∈ G such that g · x = y; 2. effective (or faithful ) if Φg = idM implies g = e; that is, g → Φg is one-to-one; and 3. free if it has no fixed points, that is, Φg (x) = x implies g = e or, equivalently, if for each x ∈ M , g → Φg (x) is one-to-one. Note that an action is free iff Gx = {e}, for all x ∈ M , and that every free action is faithful. Examples (a) Left translation Lg : G → G; h → gh, defines a transitive and free action of G on itself. Note that right multiplication Rg : G → G, h → hg, does not define a left action because Rgh = Rh ◦ Rg , so that g → Rg is an antihomomorphism. However, g → Rg does define a right action, while g → Rg−1 defines a left action of G on itself. ........................... 15 July 1998—18h02 ...........................

324. 310 9. An Introduction to Lie Groups (b) G acts on G by conjugation, g → Ig = Rg−1 ◦Lg . The map Ig : G → G given by h → ghg −1 is the inner automorphism associated with g. Orbits of this action are called conjugacy classes or, in the case of matrix groups, similarity classes. (c) Adjoint Action. Differentiating conjugation at e, we get the adjoint representation of G on g: Adg := Te Ig : Te G = g → Te G = g. Explicitly, the adjoint action of G on g is given by Ad : G × g → g, Adg (ξ) = Te (Rg−1 ◦ Lg )ξ. For example, for SO(3) we have IA (B) = ABA−1 , so differentiating with respect to B at B = identity gives AdA v = Aˆ A−1 . However, ˆ v (AdA v)(w) = Aˆ (A−1 w) = A(v × A−1 w) = Av × w, ˆ v so ˆ (AdA v) = (Av)ˆ. Identifying so(3) ∼ R3 , we get AdA v = Av. = (d) Coadjoint Action. The coadjoint action of G on g∗ , the dual of the Lie algebra g of G, is defined as follows. Let Ad∗ : g∗ → g∗ be the dual g of Adg , defined by Ad∗ α, ξ = α, Adg ξ g for α ∈ g∗ , and ξ ∈ g. Then the map Φ∗ : G × g∗ → g∗ given by (g, α) → Ad∗−1 α g is the coadjoint action of G on g∗ . The corresponding coadjoint representation of G on g∗ is denoted ∗ Ad∗ : G → GL(g∗ , g∗ ), Ad∗−1 = Te (Rg ◦ Lg−1 ) g . We will avoid the introduction of yet another ∗ by writing (Adg−1 )∗ or simply Ad∗−1 , where ∗ denotes the usual linear-algebraic dual, rather than g Ad∗ (g), in which ∗ is simply part of the name of the function Ad∗ . Any representation of G on a vector space V similarly induces a contragredient representation of G on V ∗ . ........................... 15 July 1998—18h02 ...........................

325. 9.3 Actions of Lie Groups 311 Quotient (Orbit) Spaces. An action of Φ of G on a manifold M defines an equivalence relation on M by the relation of belonging to the same orbit; explicitly, for x, y ∈ M , we write x ∼ y if there exists a g ∈ G such that g · x = y, that is, if y ∈ Orb(x) (and hence x ∈ Orb(y)). We let M/G be the set of these equivalence classes, that is, the set of orbits, sometimes called the orbit space. Let π : M → M/G : x → Orb(x), and give M/G the quotient topology by defining U ⊂ M/G to be open if and only if π −1 (U ) is open in M . To guarantee that the orbit space M/G has a smooth manifold structure, further conditions on the action are required. An action Φ : G × M → M is called proper if the mapping ˜ Φ : G × M → M × M, defined by ˜ Φ(g, x) = (x, Φ(g, x)), is proper. In finite dimensions this means that if K ⊂ M × M is compact, then Φ−1 (K) is compact. In general, this means that if {xn } is a convergent ˜ sequence in M and Φgn (xn ) converges in M , then {gn } has a convergent subsequence in G. For instance, if G is compact, this condition is automatically satisfied. Orbits of proper Lie group actions are closed and hence embedded submanifolds. The next proposition gives a useful sufficient condition for M/G to be a smooth manifold. Proposition 9.3.2. If Φ : G × M → M is a proper and free action, then M/G is a smooth manifold and π : M → M/G is a smooth submersion. For the proof, we refer to Abraham and Marsden [1978], Proposition 4.2.23. (In infinite dimensions one uses these ideas but additional technicalities often arise; see Ebin [1970] and Isenberg and Marsden [1982].) The idea of the chart construction for M/G is based on the following observation. If x ∈ M , then there is an isomorphism ϕx of Tπ(x) (M/G) with the quotient space Tx M/Tx Orb(x). Moreover, if y = Φg (x), then Tx Φg induces an isomorphism ψx,y : Tx M/Tx Orb(x) → Ty M/Ty Orb(y) satisfying ϕy ◦ ψx,y = ϕx . Examples (a) G = R acts on M = R by translations; explicitly, Φ : G × M → M, Φ(s, x) = x + s. ........................... 15 July 1998—18h02 ...........................

326. 312 9. An Introduction to Lie Groups Then for x ∈ R, Orb(x) = R. Hence M/G is a single point and the action is transitive, proper, and free. (b) G = SO(3), M = R3 (∼ so(3)∗ ). Consider the action for x ∈ R3 and = A ∈ SO(3) given by ΦA x = Ax. Then Orb(x) = {y ∈ R3 | y = x } = a sphere of radius x . Hence M/G ∼ R+ . The set = R+ = {r ∈ R | r ≥ 0} is not a manifold because it includes the endpoint r = 0. Indeed, the action is not free, since it has the fixed point 0 ∈ R3 . (c) Let G be abelian. Then Adg = idg , Ad∗−1 = idg∗ and the adjoint and g coadjoint orbits of ξ ∈ g and α ∈ g∗ , respectively, are the one-point sets {ξ} and {α}. We will see later that coadjoint orbits can be natural phase spaces for some mechanical systems like the rigid body; in particular, they are always even dimensional. Infinitesimal Generators. Next we turn to the infinitesimal description of an action, which will be a crucial concept for mechanics. Definition 9.3.3. Suppose Φ : G × M → M is an action. For ξ ∈ g, the map Φξ : R × M → M , defined by Φξ (t, x) = Φ(exp tξ, x), is an R-action on M . In other words, Φexp tξ : M → M is a flow on M . The corresponding vector field on M , given by d ξM (x) := Φexp tξ (x), dt t=0 is called the infinitesimal generator of the action corresponding to ξ. Proposition 9.3.4. The tangent space at x to an orbit Orb(x0 ) is Tx Orb(x0 ) = {ξM (x) | ξ ∈ g} , where Orb(x0 ) is endowed with the manifold structure making G/Gx0 → Orb(x0 ) into a diffeomorphism. The idea is as follows: Let σξ (t) be a curve in G tangent to ξ at t = 0. Then the map Φx,ξ (t) = Φσξ (t) (x) is a smooth curve in Orb(x0 ) with Φx,ξ (0) = x. Hence d d Φx,ξ (t) = Φσξ (t) (x) = ξM (x) dt t=0 dt t=0 ........................... 15 July 1998—18h02 ...........................

327. 9.3 Actions of Lie Groups 313 is a tangent vector at x to Orb(x0 ). Furthermore, each tangent vector is obtained in this way since tangent vectors are equivalence classes of such curves. The Lie algebra of the isotropy group Gx , x ∈ M , called the isotropy (or stabilizer , or symmetry ) algebra at x equals, by Proposition 9.1.13, gx = {ξ ∈ g | ξM (x) = 0}. Examples (a) The infinitesimal generators for the adjoint action are computed as follows. Let Ad : G × g → g, Adg (η) = Te (Rg−1 ◦ Lg )(η). For ξ ∈ g, we compute the corresponding infinitesimal generator ξg . By definition, d ξg (η) = Adexp tξ (η). dt t=0 By (9.1.5), this equals [ξ, η]. Thus, for the adjoint action, ξg = adξ ; i.e., ξg (η) = [ξ, η]. (b) We illustrate (a) for the group SO(3) as follows. Let A(t) = exp(tC), where C ∈ so(3); then A(0) = I and A (0) = C. Thus, with B ∈ so(3), d d (Adexp tC B) = (exp(tC))B(exp(tC))−1 ) dt t=0 dt t=0 d = (A(t)BA(t)−1 ) dt t=0 = A (0)BA−1 (0) + A(0)BA−1 (0). Differentiating A(t)A−1 (t) = I, we find d −1 (A (t)) = −A−1 (t)A (t)A−1 (t), dt so that A−1 (0) = −A (0) = −C. Then the preceding equation becomes d (Adexp tC B) = CB − BC = [C, B], dt t=0 as expected. ........................... 15 July 1998—18h02 ...........................

328. 314 9. An Introduction to Lie Groups (c) Let Ad∗ : G × g∗ → g∗ be the coadjoint action (g, α) → Ad∗−1 α. If g ξ ∈ g, we compute for α ∈ g∗ and η ∈ g d ξg∗ (α), η = Ad∗ exp(−tξ) (α), η dt t=0 d d = Ad∗ exp(−tξ) (α), η = α, Adexp(−tξ) η dt t=0 dt t=0 d = α, Adexp(−tξ) η dt t=0 = α, −[ξ, η] = − α, adξ (η) = − ad∗ (α), η . ξ Hence ξg∗ = − ad∗ , ξ or ξg∗ (α) = − α, [ξ, ·] . (9.3.1) , (d) Identifying so(3) ∼ (R3 , ×) and so(3)∗ ∼ R3 , using the pairing given ∗ = = by the standard Euclidean inner product, (9.3.1) reads ξso(3)∗ (l) = −l · (ξ × ·), , for l ∈ so(3)∗ and ξ ∈ so(3). For η ∈ so(3), we have ξso(3)∗ (l), η = −l · (ξ × η) = −(l × ξ) · η = − l × ξ, η , so that ξR3 (l) = −l × ξ = ξ × l. As expected, ξR3 (l) ∈ Tl Orb(l) is tangent to Orb(l) (see Figure 9.3.1). Allowing ξ to vary in so(3) ∼ R3 , one obtains all of Tl Orb(l), consistent = with Proposition 9.3.4. ξ×l l ξ Figure 9.3.1. ξR 3 (l) is tangent to Orb(l). ........................... 15 July 1998—18h02 ...........................

329. 9.3 Actions of Lie Groups 315 Equivariance. A map between two spaces is equivariant when it respects group actions on these spaces. More precisely, we state: Definition 9.3.5. Let M and N be manifolds and let G be a Lie group which acts on M by Φg : M → M , and on N by Ψg : N → N . A smooth map f : M → N is called equivariant with respect to these actions if, for all g ∈ G, f ◦ Φg = Ψg ◦ f, (9.3.2) that is, if the diagram in Figure 9.3.2 commutes. f E N M Φg Ψg c c M E N f Figure 9.3.2. Commutative diagram for equivariance. Setting g = exp(tξ) and differentiating (9.3.2) with respect to t at t = 0 gives T f ◦ ξM = ξN ◦ f . In other words, ξM and ξN are f -related. In particular, if f is an equivariant diffeomorphism, then f ∗ ξN = ξM . Also note that if M/G and N/G are both smooth manifolds with the canonical projections smooth submersions, an equivariant map f : M → N induces a smooth map fG : M/G → N/G. Averaging. A useful device for constructing invariant objects is by averaging. For example, let G be a compact group acting on a manifold M and let α be a differential form on M . Then we form α= Φ∗ α dµ(g), g G where µ is Haar measure on G. One checks that α is invariant. One can do the same with other tensors, such as Riemannian metrics on M , to obtain invariant ones. Brackets of generators. Now we come to an important formula relating the Jacobi–Lie bracket of two infinitesimal generators with the Lie algebra bracket. Proposition 9.3.6. Let the Lie group G act on the left on the manifold M . Then the infinitesimal generator map ξ → ξM of the Lie algebra g ........................... 15 July 1998—18h02 ...........................

330. 316 9. An Introduction to Lie Groups of G into the Lie algebra X(M ) of vector fields of M is a Lie algebra antihomomorphism; that is, (aξ + bη)M = aξM + bηM and [ξM , ηM ] = −[ξ, η]M , for all ξ, η ∈ g, and a, b ∈ R. To prove this, we use the following lemma: Lemma 9.3.7. (i) Let c(t) be a curve in G, c(0) = e, c (0) = ξ ∈ g. Then d ξM (x) = Φc(t) (x). dt t=0 (ii) For every g ∈ G, (Adg ξ)M = Φ∗−1 ξM . g Proof. (i) Let Φx : G → M be the map Φx (g) = Φ(g, x). Since Φx is smooth, the definition of the infinitesimal generator says that Te Φx (ξ) = ξM (x). Thus, (i) follows by the chain rule. (ii) We have d (Adg ξ)M (x) = Φ(exp(t Adg ξ), x) dt t=0 d = Φ(g(exp tξ)g −1 , x) (by Corollary 9.1.7) dt t=0 d = (Φg ◦ Φexp tξ ◦ Φg−1 (x)) dt t=0 = TΦ−1 (x) Φg ξM Φg−1 (x) g = Φ∗−1 ξM (x). g Proof of Proposition 9.3.6. Linearity follows since ξM (x) = Te Φx (ξ). To prove the second relation, put g = exp tη in (ii) of the lemma to get (Adexp tη ξ)M = Φ∗ exp(−tη) ξM . But Φexp(−tη) is the flow of −ηM , so differentiating at t = 0 the right-hand side gives [ξM , ηM ]. The derivative of the left-hand side at t = 0 equals [η, ξ]M by the preceding Example (a). ........................... 15 July 1998—18h02 ...........................

331. 9.3 Actions of Lie Groups 317 In view of this proposition one defines a left Lie algebra action of a manifold M as a Lie algebra antihomomorphism ξ ∈ g → ξM ∈ X(M ), such that the mapping (ξ, x) ∈ g × M → ξM (x) ∈ T M is smooth. Let Φ : G × G → G denote the action of G on itself by left translation: Φ(g, h) = Lg h. For ξ ∈ g, let Yξ be the corresponding right invariant vector field on G. Then ξG (g) = Yξ (g) = Te Rg (ξ), and similarly, the infinitesimal generator of right translation is the left invariant vector field g → Te Lg (ξ). Derivatives of Curves. It is convenient to have formulas for the derivatives of curves associated with the adjoint and coadjoint actions. For example, let g(t) be a (smooth) curve in G and η(t) a (smooth) curve in g. Let the action be denoted by concatenation: g(t)η(t) = Adg(t) η(t). Proposition 9.3.8. The following holds d dη g(t)η(t) = g(t) [ξ(t), η(t)] + , (9.3.3) dt dt where dg ξ(t) = g(t)−1 g(t) := Tg(t) L−1 ˙ g(t) ∈ g. dt Proof. We have d d Adg(t) η(t) = g(t0 )[g(t0 )−1 g(t)]η(t) dt t=t0 dt t=t0 d = g(t0 ) · [g(t0 )−1 g(t)]η(t) , dt t=t0 where the first g(t0 ) denotes the Ad-action, which is linear . Now g(t0 )−1 g(t) is a curve through the identity at t = t0 with tangent vector ξ(t0 ), so the above becomes dη(t0 ) g(t0 ) · [ξ(t0 ), η(t0 )] + . dt Similarly, for the coadjoint action we write g(t)µ(t) = Ad∗ −1 µ(t) g(t) and then as above, one proves that d dµ [g(t)µ(t)] = g(t) − ad∗ µ(t) + ξ(t) dt dt ........................... 15 July 1998—18h02 ...........................

332. 318 9. An Introduction to Lie Groups which we could write, extending our concatenation notation to Lie algebra actions as well, d dµ [g(t)µ(t)] = g(t) ξ(t)µ(t) + (9.3.4) dt dt where ξ(t) = g(t)−1 g(t). For right actions, these become d dη [η(t)g(t)] = η(t)ζ(t) + g(t) (9.3.5) dt dt and d dµ [µ(t)g(t)] = µ(t)ζ(t) + g(t), (9.3.6) dt dt where ζ(t) = g(t)g(t)−1 , ˙ η(t)g(t) = Adg(t)−1 η(t), and η(t)ζ(t) = −[ζ(t), η(t)] and where µ(t)g(t) = Ad∗ µ(t) g(t) and µ(t)ζ(t) = ad∗ µ(t). ζ(t) Connectivity of Some Classical Groups. First we state two facts about homogeneous spaces: 1. If H is a closed normal subgroup of the Lie group G (that is, if h ∈ H and g ∈ G, then ghg −1 ∈ H), then the quotient G/H is a Lie group and the natural projection π : G → G/H is a smooth group homomorphism. (This follows from Proposition 9.3.2; see also Varadarajan [1974] Theorem 2.9.6, p. 80.) Moreover, if H and G/H are connected then G is connected. Similarly, if H and G/H are simply connected, then G is simply connected. 2. Let G, M be ﬁnite-dimensional and second countable and let Φ : G × M → M be a transitive action of G on M and for x ∈ M , let Gx be the isotropy subgroup of x. Then the map gGx → Φg (x) is a diffeomorphism of G/Gx onto M . (This follows from Proposition 9.3.2; see also Varadarajan [1974], Theorem 2.9.4, p. 77.) The action Φ : GL(n, R) × Rn → Rn , Φ(A, x) = Ax, restricted to O(n)×S n−1 induces a transitive action. The isotropy subgroup of O(n) at en ∈ S n−1 is O(n − 1). Clearly O(n − 1) is a closed subgroup of O(n) by embedding any A ∈ O(n − 1) as ˜ A 0 A= ∈ O(n), 0 1 ........................... 15 July 1998—18h02 ...........................

333. 9.3 Actions of Lie Groups 319 and the elements of O(n−1) leave en fixed. On the other hand, if A ∈ O(n) and A(en ) = en , then A ∈ O(n − 1). It follows from 2 that the map O(n)/ O(n − 1) → S n−1 : A · O(n − 1) → A(en ) is a diffeomorphism. By a similar argument, there is a diffeomorphism S n−1 ∼ SO(n)/ SO(n − 1). = The natural action of GL(n, C) on Cn similarly induces a diffeomorphism of S 2n−1 ⊂ R2n with the homogeneous space U(n)/ U(n − 1). Moreover, we get S 2n−1 ∼ SU(n)/ SU(n − 1). In particular, since SU(1) consists only of = the 1 × 1 identity matrix, S 3 is diffeomorphic with SU(2), a fact already proved at the end of §9.2. Proposition 9.3.9. Each of the Lie groups SO(n), SU(n), and U(n) is connected for n ≥ 1, and O(n) has two components. The group SU(n) is simply connected. Proof. The groups SO(1) and SU(1) are connected since both consist only of the 1 × 1 identity matrix and U(1) is connected since U(1) = {z ∈ C | |z| = 1} = S 1 . That SO(n), SU(n), and U(n) are connected for all n now follows from fact 1 above, using induction on n and the representation of the spheres as homogeneous spaces. Since every matrix A in O(n) has determinant ±1, the orthogonal group can be written as the union of two nonempty disjoint connected open subsets as follows: O(n) = SO(n) ∪ A · SO(n), where A = diag(−1, 1, 1, . . . , 1). Thus, O(n) has two components. Here is a general strategy for proving the connectivity of the classical groups; see, for example Knapp [1996]. This works, in particular, for Sp(2m, R). Let G be a subgroup of GL(n, R) (resp. GL(n, C)) defined as the zero set of a collection of real-valued poynomials in the (real and imaginary parts) of the matrix entries. Assume, also, that G is closed under taking adjoints (see Exercise 9.2-2 for the case of Sp(2m, R)). Let K = G ∩ O(n) (resp. U (n)) and let p be the set of Hermitian matrices in g. (For Sp(2m, R), n = 2m and K = U (m); see Exercise 9.2-3). The polar decomposition says that (k, ξ) ∈ K × p → k exp(ξ) ∈ G is a homeomorphism. It follows that, since ξ lies in a connected space, G is connected iff K is connected. For Sp(2m, R) our results above show U (m) is connected, so Sp(2m, R) is connected. Tudor: where ........................... 15 July 1998—18h02 ........................... did the zero set of polys get used?

334. 320 9. An Introduction to Lie Groups Examples (a) Isometry groups. Let E be a ﬁnite-dimensional vector space with a bilinear form , . Let G be the group of isometries of E, that is, F is an isomorphism of E onto E and F e, F e = e, e , for all e, and e ∈ E. Then G is a subgroup and a closed submanifold of GL(E). The Lie algebra of G is {K ∈ L(E) | Ke, e + e, Ke = 0, for all e, e ∈ E}. (b) Lorentz group. If , denotes the Minkowski metric on R4 , that is, 3 x, y = xi y i − x4 y 4 , i=1 then the group of linear isometries is called the Lorentz group L. The dimension of L is six and L has four connected components. If I3 0 S= ∈ GL(4, R), 0 −1 then L = {A ∈ GL(4, R) | AT SA = S} and so the Lie algebra of L is Show it has 4 components? l = {A ∈ L(R4 , R4 ) | SA + AT S = 0}. The identity component of L is {A ∈ L | det A 0 and A44 0} = L+ ; ↑ L and L+ are not compact. ↑ (c) Galilean group. Consider the (closed) subgroup G of GL(5, R) that consists of matrices with the following block structure:   R v a {R, v, a, τ } :=  0 1 τ  , 0 0 1 where R ∈ SO(3), v, a ∈ R3 , and τ ∈ R. This group is called the Galilean group. Its Lie algebra is a subalgebra of L(R5 , R5 ) given by the set of matrices of the form   ˆ ω u α {ω, u, α, θ} :=  0 0 θ  , 0 0 0 ........................... 15 July 1998—18h02 ...........................

335. 9.3 Actions of Lie Groups 321 where ω, u, α ∈ R3 , and θ ∈ R. Obviously the Galilean group acts naturally on R5 ; moreover it acts naturally on R4 , embedded as the following G- invariant subset of R5 :   x x →  t , t 1 where x ∈ R3 and t ∈ R. Concretely, the action of {R, v, a, τ } on (x, t) is given by (x, t) → (Rx + tv + a, t + τ ). Thus, the Galilean group gives a change of frame of reference (unaffecting the “absolute time” variable) by rotations (R), space translations (a), time translations (τ ), and going to a moving frame, or boosts (v). We need Coadjoint Isotropy Subalgebras Are Generically Abelian (Op- to say how tional). The aim of this supplement is to prove a theorem of Duflo and simple the Vergne [1969] showing that, generically, the isotropy algebras for the coad- proof is joint action are abelian. A very simple example is G = SO(3). Here g∗ ∼ R3 = if G is and Gµ = S 1 for µ ∈ g∗ and µ = 0, and G0 = SO(3). Thus, Gµ is abelian compact. The on the open dense set g∗ {0}. proof that To prepare for the proof, we shall develop some tools. is now here If V is a finite-dimensional vector space, a subset A ⊂ V is called alge- is not very braic if it is the common zero set of a finite number of polynomial functions appealing. on V . It is easy to see that if Ai is the zero set of a finite collection of poly- E.g., one nomials Ci , for i = 1, 2, then A1 ∪ A2 is the zero set of the collection C1 C2 learns formed by all products of an element in C1 with an element in C2 . The nothing whole space V is the zero set of the constant polynomial equal to 1. Fi- about nally, if Aα is the algebraic set given as the common zeros of some finite the link collection of polynomials Cα , where α ranges over some index set, then w/maximal α Aα is the zero set of the collection α Cα . This zero set can also be tori; e.g., given as the common zeros of a finite collection of polynomials since the Gµ really zero set of any collection of polynomials coincides with the zero set of the is a torus ideal in the polynomial ring generated by this collection and any ideal in the polynomial ring over R is finitely generated (we accept this from algebra). Thus, the collection of algebraic sets in V satisfies the axioms of the collection of closed sets of a topology which is called the Zariski topology of V . Thus, the open sets of this topology are the complements of the algebraic sets. For example, the algebraic sets of R are just the finite sets, since every polynomial in R[X] has finitely many real roots (or none at all). Granting that we have a topology (the hard part), let us show that any Zariski open set in V is open and dense in the usual topology. Openness is clear, since algebraic sets are necessarily closed in the usual topology as inverse ........................... 15 July 1998—18h02 ...........................

336. 322 9. An Introduction to Lie Groups images of 0 by a continuous map. To show that a Zariski open set U is also dense, suppose the contrary, namely, that if x ∈ V U , then there is a neighborhood U1 × U2 of x in the usual topology such that (U1 × U2 ) ∩ U = ∅ and U1 ⊂ R, U2 ⊂ V2 are open, where V = R × V2 , the splitting being achieved by the choice of a basis. Since x ∈ V U , there is a finite collection of polynomials p1 , . . . , pN ∈ R[X1 , . . . , Xn ], n = dim V, that vanishes identically on U1 × U2 . If x = (x1 , . . . , xn ) ∈ V , then the polynomials qi (X1 ) = pi (X1 , x2 , . . . , xn ) ∈ R[X1 ] all vanish identically on the open set U1 ⊂ R, which is impossible since each qi has at most a finite number of roots. Therefore, (U1 × U2 ) ∩ U = ∅ is absurd and hence U must be dense in V . Theorem 9.3.10 (Duflo and Vergne [1969]). Let g be a finite-dimensional Lie algebra with dual g∗ and let r = min{dim gµ | µ ∈ g∗ }. The set {µ ∈ g∗ | dim gµ = r} is Zariski open and thus open and dense in the usual topology of g∗ . If dim gµ = r, then gµ is abelian. Proof (Due to J. Carmona, as presented in Rais [1972]). Define the map ϕµ : G → g∗ by g → Ad∗−1 µ. This is a smooth map whose range is the g coadjoint orbit Oµ through µ and whose tangent map at the identity is Te ϕµ (ξ) = − ad∗ µ. Note that ker Te ϕµ = gµ and ξ range Te ϕµ = Tµ Oµ . Thus, if n = dim g, we have rank Te ϕµ = n − dim gµ ≤ n − r since dim gµ ≥ r, for all µ ∈ g∗ . Therefore, U = {µ ∈ g∗ | dim gµ = r} = {µ ∈ g∗ | rank(Te ϕµ ) = n − r} and n − r is the maximal possible rank of all the linear maps Te ϕµ : g → g∗ , µ ∈ g∗ . Now choose a basis in g and induce the natural bases on g∗ and L(g, g∗ ). Let Si = {µ ∈ g∗ | rank Te ϕµ = n − r − i}, 1 ≤ i ≤ n − r. ........................... 15 July 1998—18h02 ...........................

337. 9.3 Actions of Lie Groups 323 Then Si is the zero set of the polynomials in µ obtained by taking all determinants of the (n − r − i + 1)-minors of the matrix representation n−r of Te ϕµ in these bases. Thus, Si is an algebraic set. Since i=1 Si is the ∗ complement of U , if follows that U is a Zariski open set in g , and hence open and dense in the usual topology of g∗ . Now let µ ∈ g∗ be such that dim gµ = r and let V be a complement to gµ in g, that is, g = V ⊕ gµ . Then Te ϕµ |V is injective. Fix ν ∈ g∗ and deﬁne S = {t ∈ R | Te ϕµ+tν |V is injective.} Note that 0 ∈ S and that S is open in R because the set of injective linear maps is open in L(g, g∗ ) and µ → Te ϕµ is continuous. Thus, S contains an open neighborhood of 0 in R. Since the rank of a linear map can only increase by slight perturbations, we have rank Te ϕµ+tν |V ≥ rank Te ϕµ |V = n − r, for |t| small, and by maximality of n − r, this forces rank Te ϕµ+tν = n − r for t in a neighborhood of 0 contained in S. Thus, for |t| small, Te ϕµ+tν |V : V → Tµ+tν Oµ+tν is an isomorphism. Hence, if ξ ∈ gµ , ad∗ (µ + tν) ∈ Tµ+tν Oµ+tν is the image ξ of a unique ξ(t) ∈ V under Te ϕµ+tν |V, that is, ξ(t) = (Te ϕµ+tν |V )−1 (ad∗ (µ + tν)). ξ This formula shows that for |t| small, t → ξ(t) is a smooth curve in V and ξ(0) = 0. However, since ad∗ (µ + tν) = −Te ϕµ+tν (ξ), ξ the deﬁnition of ξ(t) is equivalent to Te ϕµ+tν (ξ(t) + ξ) = 0, that is, ξ(t) + ξ ∈ gµ+tν . Similarly, given η ∈ gµ , there exists a unique η(t) ∈ V such that η(t) + η ∈ gµ+tν , η(0) = 0, and t → η(t) is smooth for small |t|. Therefore, the map t → µ + tν, [ξ(t) + ξ, η(t) + η] ........................... 15 July 1998—18h02 ...........................

338. 324 9. An Introduction to Lie Groups is identically zero for small |t|. In particular, its derivative at t = 0 is also zero. But this derivative equals ν, [ξ, η] + µ, [ξ (0), η] + µ, [ξ, η (0)] = ν, [ξ, η] − ad∗ µ, ξ (0) + ad∗ µ, η (0) = ν, [ξ, η] , η ξ since ξ, η ∈ gµ . Thus, ν, [ξ, η] = 0 for any ν ∈ g∗ , that is, [ξ, η] = 0. Since ξ, η ∈ gµ are arbitrary, it follows that gµ is abelian. Remarks on Infinite Dimensional Groups. We can use a slight reinterpretation of the formulae in this section to calculate the Lie algebra structure of some infinite-dimensional groups. Here we will treat this topic only formally, that is, we assume that the spaces involved are manifolds and do not specify the function space topologies. For the formal calculations, these structures are not needed, but the reader should be aware that there is a mathematical gap here. (See Ebin and Marsden [1970] and Adams, Ratiu, and Schmid [1986a,b] for more information.) Given a manifold M , let Diff(M ) denote the group of all diffeomorphisms of M . The group operation is composition. The Lie algebra of Diff(M ), as a vector space, consists of vector fields on M ; indeed the flow of a vector field is a curve in Diff(M ) and its tangent vector at t = 0 is the given vector field. To determine the Lie algebra bracket we consider the action of an arbitrary Lie group G on M . Such an action of G on M may be regarded as a homomorphism Φ : G → Diff(M ). By Proposition 9.1.5, its derivative at the identity Te Φ should be a Lie algebra homomorphism. From the definition of infinitesimal generator, we see that Te Φ · ξ = ξM . Thus, 9.1.5 suggests that [ξM , ηM ]Lie bracket = [ξ, η]M . However, by Proposition 9.3.6, [ξ, η]M = −[ξM , ηM ]. Thus, [ξM , ηM ]Lie bracket = −[ξM , ηM ]. This suggests that the Lie algebra bracket on X(M ) is minus the Jacobi–Lie bracket. ........................... 15 July 1998—18h02 ...........................

339. 9.3 Actions of Lie Groups 325 Another way to arrive at the same conclusion is to use the method of computing brackets in the table in §9.1. To do this, we first compute, according to step 1, the inner automorphism to be Iη (ϕ) = η ◦ ϕ ◦ η −1 . By step 2, we differentiate with respect to ϕ to compute the Ad map. Letting d X= ϕt , dt t=0 where ϕt is a curve in Diff(M ) with ϕ0 = Identity, we have d d Adη (X) = (Te Iη )(X) = Te Iη ϕt = Iη (ϕt ) dt t=0 dt t=0 d = (η ◦ ϕt ◦ η −1 ) = T η ◦ X ◦ η −1 = η∗ X. dt t=0 Hence Adη (X) = η∗ X. Thus, the adjoint action of Diff(M ) on its Lie algebra is just the push-forward operation on vector fields. Finally, as in step 3, we compute the bracket by differentiating Adη (X) with respect to η. But by the Lie derivative characterization of brackets and the fact that push forward is the inverse of pull back, we arrive at the same conclusion. In summary, either method suggests that: The Lie algebra bracket on Diff(M ) is minus the Jacobi–Lie bracket of vector fields. One can also say that the Jacobi–Lie bracket gives the right (as opposed to left) Lie algebra structure on Diff(M ). If one restricts to the group of volume-preserving (or symplectic) diffeomorphisms, then the Lie bracket is again minus the Jacobi–Lie bracket on the space of divergence-free (or locally Hamiltonian) vector fields. Here are three examples of actions of Diff(M ). Firstly, Diff(M ) acts on M by evaluation: the action Φ : Diff(M ) × M → M is given by Φ(ϕ, x) = ϕ(x). Secondly, the calculations we did for Adη show that the adjoint action of Diff(M ) on its Lie algebra is given by push forward. Thirdly, if we identify the dual space X(M )∗ with one-form densities by means of integration, then the change of variables formula shows that the coadjoint action is given by push forward of one-form densities. ........................... 15 July 1998—18h02 ...........................

340. 326 9. An Introduction to Lie Groups (d) Unitary Group of Hilbert Space. Another basic example of an infinite-dimensional group is the unitary group U(H) of a complex Hilbert space H. If G is a Lie group and ρ : G → U(H) is a group homomorphism, we call ρ a unitary representation. In other words, ρ is an action of G on H by unitary maps. As with the diffeomorphism group, questions of smoothness regarding U(H) need to be dealt with carefully and in this book we shall only give a brief indication of what is involved. The reason for care is, for one thing, because one ultimately is dealing with PDE’s rather than ODE’s and the hypotheses made must be such that PDE’s are not excluded. For example, for a unitary representation one assumes that for each ψ, ϕ ∈ H, the map g → ψ, ρ(g)ϕ of G to C is continuous. In particular, for G = R one has the notion of a continuous one-parameter group U (t) so that U (0) = identity and U (t + s) = U (t) ◦ U (s). Stone’s theorem says that in an appropriate sense we can write U (t) = etA where A is an (unbounded) skew-adjoint operator defined on a dense domain D(A) ⊂ H. See, for example, Abraham, Marsden and Ratiu [1988, §7.4B] for the proof. Conversely each skew-adjoint operator defines a one parameter subgroup. Thus, Stone’s theorem gives precise meaning to the statement: the Lie algebra u(H) of U(H) consists of the skew adjoint operators. The Lie bracket is the commutator, as long as one is careful with domains. If ρ is a unitary representation of a finite dimensional Lie group G on H, then ρ(exp(tξ)) is a one-parameter subgroup of U(H), so Stone’s theorem guarantees that there is a map ξ → A(ξ) associating a skew-adjoint operator A(ξ) to each ξ ∈ g. Formally we have [A(ξ), A(η)] = [ξ, η]. Results like this are aided by a theorem of Nelson [1959] guaranteeing a dense subspace DG ⊂ H such that (i) A(ξ) is well-defined on DG , (ii) A(ξ) maps DG to DG , and (iii) for ψ ∈ DG , [exp tA(ξ)]ψ is C ∞ in t with derivative at t = 0 given by A(ξ)ψ. ........................... 15 July 1998—18h02 ...........................

341. 9.3 Actions of Lie Groups 327 This space is called an essential G-smooth part of H and on DG the above commutator relation and the linearity A(αξ + βη) = αA(ξ) + βA(η) become literally true. Moreover, we loose little by using DG since A(ξ) is uniquely determined by what it is on DG . We identify U(1) with the unit circle in C and each such complex number determines an element of U(H) by multiplication. Thus, we regard U(1) ⊂ U(H). As such, it is a normal subgroup (in fact, elements of U(1) commute with elements of U(H)), so the quotient is a group called the projective unitary group of H. We write it as U(PH) = U(H)/ U(1). We write elements of U(PH) as [U ] regarded as an equivalence class of U ∈ U(H). The group U(PH) acts on projective Hilbert space PH = H/C, as in §5.3, by [U ][ϕ] = [U ϕ]. One parameter subgroups of U(PH) are of the form [U (t)] for a one parameter subgroup U (t) of U(H). This is a particularly simple case of the general problem considered by Bargmann and Wigner of lifting projective representations, a topic we return to later. In any case, this means we can identify the Lie algebra as u(PH) = u(H)/iR, where we identify the two skew adjoint operators A and A + λi, for λ real. A projective representation of a group G is a homomorphism τ : G → U(PH); we require continuity of | ψ, τ (g)ϕ |, which is well deﬁned for [ψ], [ϕ] ∈ PH. There is an analogue of Nelson’s theorem that guarantees an essential G-smooth part PDG of PH with properties like those of DG . Exercises 9.3-1. Let a Lie group G act linearly on a vector space V . Deﬁne a group structure on G × V by (g1 , v1 ) · (g2 , v2 ) = (g1 g2 , g1 v2 + v1 ). Show that this makes G × V into a Lie group—it is called the semidirect product and is denoted G V . Determine its Lie algebra g V . 9.3-2. ........................... 15 July 1998—18h02 ...........................

342. 328 9. An Introduction to Lie Groups (a) Show that the Euclidean group E(3) can be written as O(3) R3 in the sense of the preceding exercise. (b) Show that E(3) is isomorphic to the group of (4 × 4)-matrices of the form A b , 0 1 where A ∈ O(3) and b ∈ R3 . 9.3-3. Show that the Galilean group is a semidirect product G = (SO(3) R3 ) R4 . Compute explicitly the inverse of a group element, the adjoint and the coadjoint actions. 9.3-4. If G is a Lie group, show that T G is isomorphic (as a Lie group) with G g (see Exercise 9.1-2). Link w/ 9.3-5. In the Relative Darboux Theorem of Exercise 5.1-5, assume that equivariant a compact Lie group G acts on P , that S is a G-invariant submanifold and Darboux. that both Ω0 and Ω1 are G-invariant. Conclude that the diffeomorphism ϕ : U −→ ϕ(U ) can be chosen to commute with the G-action and that V , ϕ(U ) can be chose to be a G-invariant. 9.3-6. Verify, using standard vector notation, the four “derivative of curves” formulae for SO(3). 9.3-7. Prove the following generalization of the Duflo–Vergne Theorem due to Guillemin and Sternberg [1984]. Let S be an infinitesimally invariant submanifold of g∗ , that is, ad∗ µ ∈ S, whenever µ ∈ S and ξ ∈ g. Let ξ r = min{dim gµ |µ ∈ S}. Then dim gµ = r implies [gµ , gµ ] ⊂ (Tµ S)0 = {ξ ∈ g | u, ξ = 0, for all u ∈ Tµ S}. In particular gµ /(Tµ S)0 is abelian. (The Duflo–Vergne Theorem is the case for which S = g∗ .) 9.3-8. Use the Complex Polar Decomposition Theorem 9.2.15 and simple connectedness of SU(n) to show that SL(n, C) is also simply connected. 9.3-9. Show that SL(2, C) is the simply connected covering group of the identity component L† of the Lorentz group. ↑ ........................... 15 July 1998—18h02 ...........................

343. This is page 329 Printer: Opaque this 10 Poisson Manifolds The dual g∗ of a Lie algebra g carries a Poisson bracket given by δF δG {F, G} (µ) = µ, , δµ δµ for µ ∈ g∗ , a formula found by Lie [1890], §75. As we saw in the Introduc- tion, this Lie–Poisson bracket plays an important role in the Hamiltonian description of many physical systems. This bracket is not the bracket associated with any symplectic structure on g∗ , but is an example of the more general concept of a Poisson manifold . However, the Lie–Poisson bracket is associated with a symplectic structure on coadjoint orbits and with the canonical symplectic structure on T ∗ G. These facts are developed in Chap- ters 13 and 14. Chapter 15 shows how this works in detail for the rigid body. 10.1 The Deﬁnition of Poisson Manifolds This section generalizes the notion of a symplectic manifold by keeping just enough of the properties of Poisson brackets to describe Hamiltonian systems. The history of Poisson manifolds is complicated by the fact that the notion was rediscovered many times under diﬀerent names; they occur in the works of Lie [1890], Dirac [1930], [1964], Pauli [1953], Martin [1959], Jost [1964], Arens [1970], Hermann [1973], Sudarshan and Mukunda [1974], Vinogradov and Krasilshchik [1975], and Lichnerowicz [1975b]. The name Poisson manifold was coined by Lichnerowicz. Further historical comments are given in §10.3

344. 330 10. Poisson Manifolds Definition 10.1.1. A Poisson bracket (or a Poisson structure) on a manifold P is a bilinear operation { , } on F(P ) = C ∞ (P ) such that: (i) (F(P ), { , }) is a Lie algebra; and (ii) { , } is a derivation in each factor, that is, {F G, H} = {F, H} G + F {G, H} , for all F, G, and H ∈ F(P ). A manifold P endowed with a Poisson bracket on F(P ) is called a Poisson manifold. A Poisson manifold is denoted by (P, { , }) or simply by P if there is no danger of confusion. Note that any manifold has the trivial Poisson structure which is defined by setting {F, G} = 0, for all F, G ∈ F(P ). Occasionally we consider two different Poisson brackets { , }1 and { , }2 on the same manifold; the two distinct Poisson manifolds are then denoted by (P, { , }1 ) and (P, { , }2 ). The notation { , }P for the bracket on P is also used when confusion might arise. Examples (a) Symplectic Bracket. Any symplectic manifold is a Poisson manifold . The Poisson bracket is defined by the symplectic form as was shown in §5.5. Condition (ii) of the definition is satisfied as a consequence of the derivation property of vector fields: {F G, H} = XH [F G] = F XH [G] + GXH [F ] = F {G, H} + G{F, H}. (b) Lie–Poisson Bracket. If g is a Lie algebra, then its dual g∗ is a Poisson manifold with respect to each of the Lie–Poisson brackets { , }+ and { , }− defined by δF δG {F, G}± (µ) = ± µ, , (10.1.1) δµ δµ for µ ∈ g∗ and F, G ∈ F(g∗ ). The properties of a Poisson bracket can be easily verified. Bilinearity and skew-symmetry are obvious. The derivation property of the bracket follows from the Leibniz rule for functional derivatives δ(F G) δG δF = F (µ) + G(µ). δµ δµ δµ ........................... 15 July 1998—18h02 ...........................

345. 10.1 The Definition of Poisson Manifolds 331 The Jacobi identity for the Lie–Poisson bracket follows from the Jacobi identity for the Lie algebra bracket and the formula δ δF δG ± {F, G}± = , − D2 F (µ) ad∗ δG/δµ µ, · δµ δµ δµ + D2 G(µ) ad∗ δF/δµ µ, · , (10.1.2) where we recall from the preceding chapter that for each ξ ∈ g, adξ : g → g denotes the map adξ (η) = [ξ, η] and ad∗ : g∗ → g∗ is its dual. We give a ξ different proof that (10.1.1) is a Poisson bracket in Chapter 13. (c) Rigid Body Bracket. Specializing Example (b) to the Lie algebra of the rotation group, so(3) ∼ R3 , and identifying R3 and (R3 )∗ via the = standard inner product, we get the following Poisson structure on R3 : {F, G}− (Π) = −Π · ( F × G), (10.1.3) where Π ∈ R3 and F , the gradient of F , is evaluated at Π. The Poisson bracket properties can be verified by direct computation in this case; see Exercise 1.2-1. We call (10.1.3) the rigid body bracket. (d) Ideal Fluid Bracket. Specialize the Lie–Poisson bracket to the Lie algebra Xdiv (Ω) of divergence-free vector fields defined in a region Ω of R3 and tangent to ∂Ω, with the Lie bracket being the negative of the Jacobi– Lie bracket. Identify X∗ (Ω) with Xdiv (Ω) using the L2 pairing div v, w = v · w d3 x, (10.1.4) Ω where v · w is the ordinary dot product in R3 . Thus, the plus Lie–Poisson bracket is δF δG 3 {F, G}(v) = − v· , d x, (10.1.5) Ω δv δv where the functional derivative δF/δv is the element of Xdiv (Ω) defined by 1 δF lim [F (v + εδv) − F (v)] = · δv d3 x. ε→0 ε Ω δv (e) Poisson–Vlasov Bracket. Let (P, { , }P ) be a Poisson manifold and let F(P ) be the Lie algebra of functions under the Poisson bracket. Iden- tify F(P )∗ with densities f on P . Then the Lie–Poisson bracket has the expression δF δG {F, G}(f ) = f , . (10.1.6) P δf δf P ........................... 15 July 1998—18h02 ...........................

346. 332 10. Poisson Manifolds (f ) Frozen Lie–Poisson Bracket. Fix (or “freeze”) ν ∈ g∗ and define for any F, G ∈ F(g∗ ) the bracket δF δG {F, G}ν (µ) = ± ν, ± , . (10.1.7) δµ δµ The properties of a Poisson bracket are verified as in the case of the Lie–Poisson bracket, the only difference being that (10.1.2) is replaced by δ ± {F, G}ν = −D2 F (ν) ad∗ ± ∗ δG/δµ µ, · + D G(ν) adδF/δµ µ, · 2 δµ (10.1.8) This bracket is useful in the description of the Lie–Poisson equations linearized at an equilibrium point.1 (g) KdV Bracket. Let S = [S ij ] be a symmetric matrix. On F(Rn , Rn ), set ∞ n δF d δG d δG δF {F, G}(u) = S ij − dx (10.1.9) −∞ i,j=1 δui dx δuj dx δuj δui for functions F, G satisfying δF/δu, and δG/δu → 0 as x → ±∞. This is a Poisson structure that is useful for the KdV equation and for gas dynamics (see Benjamin [1984]).2 If S is invertible and S −1 = [Sij ], then (10.1.9) is the Poisson bracket associated with the weak symplectic form ∞ n y Ω(u, v) = 1 2 Sij ui (x) dx v j (y) −∞ i,j=l −∞ y − v j (x) dx ui (y) dy. (10.1.10) −∞ This is easily seen by noting that XH (u) is given by d δH XH (u) = S ij i . dx δuj (h) Toda Lattice Bracket. Let P = (a, b) ∈ R2n | ai 0, i = 1, . . . , n 1 See, for example, Abarbanel, Holm, Marsden, and Ratiu [1986]. 2 Thisis a particular case of Example (f), the Lie algebra being the pseudo-differential operators on the line of order ≤ −1 and ν = dS/dx. ........................... 15 July 1998—18h02 ...........................

347. 10.1 The Definition of Poisson Manifolds 333 and consider the bracket   ∂G T T ∂F ∂F   {F, G}(a, b) = , W  ∂a  , (10.1.11) ∂a ∂b ∂G ∂b T where (∂F/∂a) is the row vector ∂F/∂a1 , . . . , ∂F/∂an , etc., and  1  a 0 0 A  ..  W= , where A =  . . (10.1.12) −A 0 0 an In terms of the coordinate functions ai , bj , the bracket (10.1.11) is given by ai , aj = 0, bi , bj = 0, (10.1.13) ai , bj = 0 if i = j, i j i a ,b =a if i = j. This Poisson bracket is determined by the symplectic form n 1 i Ω=− da ∧ dbi (10.1.14) i=1 ai as an easy verification shows. The mapping (a, b) → (log a−1 , b) is a symplectic diffeomorphism of P with R2n endowed with the canonical symplectic structure. This symplectic structure is known as the first Poisson structure of the non-periodic Toda lattice. We shall not study this example in any detail in this book, but we point out that its bracket is the restriction of a Lie–Poisson bracket to a certain coadjoint orbit of the group of lower triangular matrices; we refer the interested reader to §14.5, Kostant [1979], and Symes [1980, 1982a,b] for further information. Exercises 10.1-1. If P1 and P2 are Poisson manifolds, show how to make P1 × P2 into a Poisson manifold. 10.1-2. Verify directly that the Lie–Poisson bracket satisfies Jacobi’s identity. 10.1-3 (A Quadratic Bracket). Let A = Aij be a skew-symmetric matrix. On Rn , define B ij = Aij xi xj (no sum). Show that the following defines a Poisson structure: n ∂F ∂G {F, G} = B ij . i,j=1 ∂xi ∂xj ........................... 15 July 1998—18h02 ...........................

348. 334 10. Poisson Manifolds 10.1-4 (A Cubic Bracket). For x = (x1 , x2 , x3 ) ∈ R3 , put x1 , x2 = x 2 x3 , {x2 , x3 } = x 2 x1 , {x3 , x1 } = x 2 x2 . Let B ij = xi , xj , for i j and i, j = 1, 2, 3, set B ji = −B ij , and define n ∂F ∂G {F, G} = B ij . i,j=1 ∂xi ∂xj Check that this makes R3 into a Poisson manifold. 10.1-5. Let Φ : g∗ → g∗ be a smooth function and define for F, H : g∗ → R, δF δH {F, H}Φ (µ) = Φ(µ), , . δµ δµ (a) Show that this rule defines a Poisson bracket on g∗ if and only if Φ satisfies the following identity. DΦ(µ) · ad∗ (µ), [η, ξ] + DΦ(µ) · ad∗ Φ(µ), [ξ, ζ] ζ η + DΦ(µ) · ad∗ Φ(µ), [ζ, η] = 0, ξ for all ξ, η, ζ ∈ g, and all µ ∈ g∗ . (b) Show that this relation holds if Φ(µ) = µ and Φ(µ) = ν, a fixed element of g∗ , thereby obtaining the Lie–Poisson structure (10.1.1) and the linearized Lie–Poisson structure (10.1.7) on g∗ . Show that it also holds if Φ(µ) = aµ + ν for some a ∈ R. (c) Assume g has a weakly nondegenerate invariant bilinear form κ : g × g → R and identify g∗ with g by κ. If Ψ : g → g is smooth, show that {F, H}Ψ (ξ) = κ(Ψ(ξ), [ F (ξ), H(ξ)]) is a Poisson bracket if and only if κ(DΨ(λ) · [Ψ(λ), ζ], [η, ξ]) + κ(DΨ(λ) · [Ψ(λ), η], [ξ, ζ]) + κ(DΨ(λ) · [Ψ(λ), ξ], [ζ, η]) = 0, for all λ, ξ, η, ζ ∈ g. Here, F (ξ), H(ξ) ∈ g are the gradients of F and H at ξ ∈ g relative to κ. Conclude as in (b) that this relation holds if Ψ(λ) = aλ + χ for a ∈ R and χ ∈ g. ........................... 15 July 1998—18h02 ...........................

349. 10.2 Hamiltonian Vector Fields and Casimir Functions 335 (d) In the hypothesis of (c), let Ψ(λ) = ψ(λ) for some smooth ψ : g → R. Show that { , }Ψ is a Poisson bracket if and only if D2 ψ(λ)([ ψ(λ), ζ], [η, ξ]) − D2 ψ(λ)( ψ(λ), [ζ, [η, ξ]]) + D2 ψ(λ)([ ψ(λ), η], [ξ, ζ]) − D2 ψ(λ)( ψ(λ), [η, [ξ, ζ]]) + D2 ψ(λ)([ ψ(λ), ξ], [ζ, η]) − D2 ψ(λ)( ψ(λ), [ξ, [ζ, η]]) = 0, for all λ, ξ, η, ζ ∈ g. In particular, if D2 ψ(λ) is an invariant bilinear form for all λ, this condition holds. However, if g = so(3) and ψ is arbitrary, then this condition also holds (see Exercise 1.3-2.) 10.2 Hamiltonian Vector Fields and Casimir Functions Hamiltonian Vector Fields. We begin by extending the notion of a Hamiltonian vector field from the symplectic to the Poisson context. Proposition 10.2.1. Let P be a Poisson manifold. If H ∈ F(P ), then there is a unique vector field XH on P such that XH [G] = {G, H}, (10.2.1) for all G ∈ F(P ). We call XH the Hamiltonian vector field of H. Proof. This is a consequence of the fact that any derivation on F(P ) is represented by a vector field. Fixing H, the map G → {G, H} is a derivation, and so it uniquely determines XH satisfying (10.3.1). (In infinite dimensions some technical conditions are needed for this proof, which are deliberately ignored here; see Abraham, Marsden, and Ratiu [1988], §4.2.) Notice that (10.2.1) agrees with our definition of Poisson brackets in the symplectic case, so if the Poisson manifold P is symplectic, XH defined here agrees with the definition in §5.5. Proposition 10.2.2. The map H → XH of F(P ) to X(P ) is a Lie algebra antihomomorphism; that is, [XH , XK ] = −X{H,K} . Proof. Using Jacobi’s identity, we find that [XH , XK ][F ] = XH [XK [F ]] − XK [XH [F ]] = {{F, K} , H} − {{F, H} , K} = − {F, {H, K}} = − X{H,K} [F ]. ........................... 15 July 1998—18h02 ...........................

350. 336 10. Poisson Manifolds Poisson Bracket Form. ˙ Next, we establish the equation F = {F, H} in the Poisson context. Proposition 10.2.3. Let ϕ be a flow on a Poisson manifold P . Then (i) for any F ∈ F(U ), U open in P , d (F ◦ ϕt ) = {F, H} ◦ ϕt = {F ◦ ϕt , H}, dt or, for short, ˙ F = {F, H}, for any F ∈ F(U ), U open in P , if and only if ϕt is the flow of XH . (ii) If ϕt is the flow of XH , then H ◦ ϕt = H. Proof. (i) Let z ∈ P . Then d d F (ϕt (z)) = dF (ϕt (z)) · ϕt (z) dt dt and {F, H}(ϕt (z)) = dF (ϕt (z)) · XH (ϕt (z)). The two expressions are equal for any F ∈ F(U ), U open in P , if and only if d ϕt (z) = XH (ϕt (z)), dt by the Hahn–Banach theorem. This is equivalent to t −→ ϕt (z) being the integral curve of XH with initial condition z, that is, ϕt is the flow of XH . On the other hand, if ϕt is the flow of XH , then we have XH (ϕt (z)) = Tz ϕt (XH (z)) so that by the chain rule d F (ϕt (z)) = dF (ϕt (z)) · XH (ϕt (z)) dt = dF (ϕt (z)) · Tz ϕt (XH (z)) = d(F ◦ ϕt )(z) · XH (z) = {F ◦ ϕt , H}(z). (ii) For the proof of (ii), let H = F in (i). Corollary 10.2.4. Let G, H ∈ F(P ). Then G is constant along the integral curves of XH if and only if {G, H} = 0, if and only if H is constant along the integral curves of XG . ........................... 15 July 1998—18h02 ...........................

351. 10.2 Hamiltonian Vector Fields and Casimir Functions 337 Among the elements of F(P ) are functions C such that {C, F } = 0, for all F ∈ F(P ), that is, C is constant along the flow of all Hamiltonian vector fields or, equivalently, XC = 0, that is, C generates trivial dynamics. Such functions are called Casimir functions of the Poisson structure. They form the center of the Poisson algebra.3 This terminology is used in, for example, Sudarshan and Mukunda [1974]. H. B. G. Casimir is a prominent physicist who wrote his thesis (Casimir [1931]) on the quantum mechanics of the rigid body, under the direction of Paul Ehrenfest. Recall that it was Ehrenfest who, in his thesis, worked on the variational structure of ideal flow in Lagrangian or material representation. Some History of Poisson Structures.4 Following from the work of Lagrange and Poisson discussed at the end of §8.1, the general concept of a Poisson manifold should be credited to Sophus Lie in his treatise on transformation groups written around 1880 in the chapter on “function groups.” Lie uses the word “group” for both “group” and “algebra.” For example, a “function group” should really be translated as “function algebra.” On page 237, Lie defines what today is called a Poisson structure. The title of Chapter 19 is The Coadjoint Group, which is explicitly identified on page 334. Chapter 17, pages 294-298, defines a linear Poisson structure on the dual of a Lie algebra, today called the Lie–Poisson structure, and “Lie’s Third Theorem” is proved for the set of regular elements. On page 349, together with a remark on page 367, it is shown that the Lie–Poisson structure naturally induces a symplectic structure on each coadjoint orbit. As we shall point out in §11.2, Lie also had many of the ideas of momentum maps. For many years this work appears to have been forgotten. Because of the above history, Marsden and Weinstein [1983] coined the phrase “Lie–Poisson bracket” for this object, and this terminology is now in common use. However, it is not clear that Lie understood the fact that the Lie–Poisson bracket is obtained by a simple reduction process, namely, that it is induced from the canonical cotangent Poisson bracket on T ∗ G by passing to g∗ regarded as the quotient T ∗ G/G, as will be explained in Chapter 13. The link between the closedness of the symplectic form and the Jacobi identity is a little harder to trace explicitly; some comments in this direction are given in Souriau [1970], who gives credit to Maxwell. Lie’s work starts by taking functions F1 , . . . , Fr on a symplectic manifold M , with the property that there exist functions Gij of r variables, such that {Fi , Fj } = Gij (F1 , . . . , Fr ). 3 The center of a group (or algebra) is the set of elements that commute with all elements of the group (or algebra). 4 We thank Hans Duistermaat and Alan Weinstein for their help with the comments in this section; the paper of Weinstein [1983a] should also be consulted by the interested reader. ........................... 15 July 1998—18h02 ...........................

352. 338 10. Poisson Manifolds In Lie’s time, all functions in sight are implicitly assumed to be analytic. The collection of all functions φ of F1 , . . . , Fr is the “function group”; it is provided with the bracket [φ, ψ] = Gij φi ψj , (10.2.2) ij where ∂φ ∂ψ φi = and ψj = . ∂Fi ∂Fj Considering F = (F1 , . . . , Fr ) as a map from M to an r-dimensional space P , and φ and ψ as functions on P , one may formulate this as: [φ, ψ] is a Poisson structure on P , with the property that F ∗ [φ, ψ] = {F ∗ φ, F ∗ ψ}. Lie writes down the equations for the Gij that follow from the antisymmetry and the Jacobi identity for the bracket { , } on M . He continues with the question: if a given a system of functions Gij in r variables satisfy these equations, is it induced, as above, from a function group of functions of 2n variables? He shows that under suitable rank conditions the answer is yes. As we shall see below, this result is the precursor to many of the fundamental results about the geometry of Poisson manifolds. It is obvious that if Gij is a system that satisﬁes the equations that Lie writes down, then (10.2.2) is a Poisson structure in r-dimensional space. Vice versa, for any Poisson structure [φ, ψ], the functions Gij = [Fi , Fj ] satisfy Lie’s equations. Lie continues with more remarks on local normal forms of function groups, (i.e., of Poisson structures), under suitable rank conditions, which are not always stated as explicitly as one would like. These amount to the following: a Poisson structure of constant rank is the same as a foliation with symplectic leaves. It is this characterization that Lie uses to get the symplectic form on the coadjoint orbits. On the other hand, Lie does not apply the symplectic form on the coadjoint orbits to representation theory. Representation theory of Lie groups started only later with Schur on GL(n), and was continued by Elie Cartan with representations of semisimple Lie algebras, and in the 1930s, by Weyl with the representation of compact Lie groups. The coadjoint orbit symplectic structure was connected with representation theory in the work of Kirillov and Kostant. On the other hand, Lie did apply the Poisson structure on the dual of the Lie algebra to prove that every abstract Lie algebra can be realized as a Lie algebra of Hamiltonian vector ﬁelds, or as a Lie subalgebra of the Poisson algebra ........................... 15 July 1998—18h02 ...........................

353. 10.2 Hamiltonian Vector Fields and Casimir Functions 339 of functions on some symplectic manifold. This is “Lie’s third fundamental theorem” in the form given by Lie. Of course, in geometry, people like Engel, Study and, in particular, Elie Cartan, studied Lie’s work intensely and propagated it very actively. How- ever, through the tainted glasses of retrospection, Lie’s work on Poisson structures did not appear to receive as much attention in mechanics as it deserved; for example, even though Cartan himself did very important work in mechanics (such as, Cartan [1923, 1928a,b]), he did not seem to realize that the Lie–Poisson bracket was central to the Hamiltonian description of some of the rotating ﬂuid systems he was studying. However, others, such as Hamel [1904, 1949], did study Lie intensively and used it to make substantial contributions and extensions (such as to the study of nonholonomic systems, including rolling constraints), but many other active schools seem to have missed it. Even more surprising in this context is the contribution of Poincar´ [1901b, 1910] to the Lagrangian side of the e story, a tale to which we shall come in Chapter 13. Examples (a) Symplectic Case. On a symplectic manifold P , any Casimir function is constant on connected components of P . This holds since in the symplectic case, XC = 0 implies dC = 0 and hence C is locally constant. (b) Rigid Body Casimirs. In the context of Example (c) of §10.1, let C(Π) = Π 2 /2. Then C(Π) = Π and by the properties of the triple product, we have for any F ∈ F(R3 ), {C, F } (Π) = −Π · ( C × F ) = − Π · (Π × F) = − F · (Π × Π) = 0. This shows that C(Π) = Π 2 /2 is a Casimir function. A similar argument shows that 1 2 CΦ (Π) = Φ 2 Π (10.2.3) is a Casimir function, where Φ is an arbitrary (diﬀerentiable) function of one variable; this is proved by noting that 1 2 CΦ (Π) = Φ 2 Π Π. (c) Helicity. In Example (d) of §10.1, the helicity C(v) = v·( × v) d3 x (10.2.4) Ω can be checked to be a Casimir function if ∂Ω = ∅. ........................... 15 July 1998—18h02 ...........................

354. 340 10. Poisson Manifolds (d) Poisson–Vlasov Casimirs. In Example (e) of §10.1, given a differentiable function Φ : R → R, the map C : F(P ) → R defined by C(f ) = Φ(f (q, p)) dq dp (10.2.5) is a Casimir function. Here we choose P to be symplectic, have written dq dp = dz for the Liouville measure, and have used it to identify functions and densities. Exercises 10.2-1. Verify the relation [XH , XK ] = −X{H,K} directly for the rigid body bracket. 10.2-2. Verify that (10.2.5): C(f ) = Φ(f (q, p)) dq dp, defines a Casimir function. 10.2-3. Let P be a Poisson manifold and let M ⊂ P be a connected submanifold with the property that for each v ∈ Tx M there is a Hamiltonian vector field XH on P such that v = XH (x); that is, Tx M is spanned by Hamiltonian vector fields. Prove that any Casimir function is constant on M. 10.3 Properties of Hamiltonian Flows Hamiltonian Flows Are Poisson. Now we establish the Poisson analog of the symplectic nature of the flows of Hamiltonian vector fields. Proposition 10.3.1. If ϕt is the flow of XH , then ϕ∗ {F, G} = {ϕ∗ F, ϕ∗ G} ; t t t in other words, {F, G} ◦ ϕt = {F ◦ ϕt , G ◦ ϕt } . Thus, the flows of Hamiltonian vector fields preserve the Poisson structure. Proof. This is actually true even for time-dependent Hamiltonian systems (as we will see later), but here we will prove it only in the time- independent case. Let F, K ∈ F(P ) and let ϕt be the flow of XH . Let u = {F ◦ ϕt , K ◦ ϕt } − {F, K} ◦ ϕt . ........................... 15 July 1998—18h02 ...........................

355. 10.3 Properties of Hamiltonian Flows 341 Because of the bilinearity of the Poisson bracket, du d d d = F ◦ ϕt , K ◦ ϕ t + F ◦ ϕt , K ◦ ϕt − {F, K} ◦ ϕt . dt dt dt dt Using Proposition 10.2.3, this becomes du = {{F ◦ ϕt , H} , K ◦ ϕt } + {F ◦ ϕt , {K ◦ ϕt , H}} − {{F, K} ◦ ϕt , H} , dt which, by Jacobi’s identity, gives du = {u, H} = XH [u]. dt The unique solution of this equation is ut = u0 ◦ ϕt . Since u0 = 0, we get u = 0, which is the result. As in the symplectic case, with which this is of course consistent, this argument shows how Jacobi’s identity plays a crucial role. Poisson Maps. A smooth mapping f : P1 → P2 between the two Poisson manifolds (P1 , { , }1 ) and (P2 , { , }2 ) is called canonical or Poisson if f ∗ {F, G}2 = {f ∗ F, f ∗ G}1 , for all F, G ∈ F(P2 ). Proposition 10.3.1 shows that flows of Hamiltonian vector fields are canonical maps. We saw already in Chapter 5 that if P1 and P2 are symplectic manifolds, a map f : P1 → P2 is canonical if and only if it is symplectic. Properties of Poisson Maps. The next proposition shows that Poisson maps push Hamiltonian flows to Hamiltonian flows. Proposition 10.3.2. Let f : P1 → P2 be a Poisson map and let H ∈ F(P2 ). If ϕt is the flow of XH and ψt is the flow of XH◦f , then ϕ t ◦ f = f ◦ ψt and T f ◦ XH◦f = XH ◦ f. Conversely, if f is a map from P1 to P2 and for any H ∈ F(P2 ), the Hamiltonian vector fields XH◦f ∈ X(P1 ) and XH ∈ X(P2 ) are f -related, that is, T f ◦ XH◦f = XH ◦ f, then f is canonical. Proof. For any G ∈ F(P2 ) and z ∈ P1 , Proposition 10.2.3(i) and the definition of Poisson maps yield d d G((f ◦ ψt )(z)) = (G ◦ f )(ψt (z)) dt dt = {G ◦ f, H ◦ f } (ψt (z)) = {G, H} (f ◦ ψt )(z), ........................... 15 July 1998—18h02 ...........................

356. 342 10. Poisson Manifolds that is, (f ◦ ψt )(z) is an integral curve of XH on P2 through the point f (z). Since (ϕt ◦f )(z) is another such curve, uniqueness of integral curves implies that (f ◦ ψt )(z) = (ϕt ◦ f )(z). The relation T f ◦ XH◦f = XH ◦ f follows from f ◦ ψt = ϕt ◦ f by taking the time-derivative. Conversely, assume that for any H ∈ F(P2 ) we have T f ◦XH◦f = XH ◦f . Therefore, by the chain rule, XH◦f [F ◦ f ] (z) = dF (f (z)) · Tz f (XH◦f (z)) = dF (f (z)) · XH (f (z)) = XH [F ] (f (z)), that is, XH◦f [f ∗ F ] = f ∗ (XH [F ]). Thus, for G ∈ F(P2 ), {G, H} ◦ f = f ∗ (XH [G]) = XH◦f [f ∗ G] = {G ◦ f, H ◦ f } and so f is canonical. Exercises 10.3-1. Verify directly that a rotation R : R3 → R3 is a Poisson map for the rigid body bracket. 10.3-2. If P1 and P2 are Poisson manifolds, show that the projection π1 : P1 × P2 → P1 is a Poisson map. Is the corresponding statement true for symplectic maps? 10.4 The Poisson Tensor Deﬁnition of the Poisson Tensor. By the derivation property of the Poisson bracket, the value of the bracket {F, G} at z ∈ P (and thus XF (z) as well), depends on F only through dF (z) (see Abraham, Marsden, and Ratiu [1988], Theorem 4.2.16 for this type of argument). Thus, there is a contravariant antisymmetric two-tensor B : T ∗P × T ∗P → R such that B(z)(αz , βz ) = {F, G} (z), ∗ where dF (z) = αz and dG(z) = βz ∈ Tz P . This tensor B is called a cosymplectic or Poisson structure. In local coordinates (z 1 , . . . , z n ), B is determined by its matrix elements z I , z J = B IJ (z) and the bracket becomes ∂F ∂G {F, G} = B IJ (z) . (10.4.1) ∂z I ∂z J ........................... 15 July 1998—18h02 ...........................

357. 10.4 The Poisson Tensor 343 Let B : T ∗ P → T P be the vector bundle map associated to B, that is, B(z)(αz , βz ) = αz , B (z)(βz ) . ˙ Consistent with our conventions F = {F, H}, the Hamiltonian vector ˙ field is given by XH (z) = Bz · dH(z). Indeed, F (z) = dF (z) · XH (z) and {F, H} (z) = B(z)(dF (z), dH(z)) = dF (z), B (z)(dH(z)) . Comparing these expressions gives the stated result. Coordinate Representation. A convenient way to specify a bracket in finite dimensions is by giving the coordinate relations z I , z J = B IJ (z). The Jacobi identity is then implied by the special cases zI , zJ , zK + zK , zI , zJ + z J , z K , z I = 0, which are equivalent to the differential equations ∂B JK ∂B KI ∂B IJ B LI + B LJ + B LK =0 (10.4.2) ∂z L ∂z L ∂z L (the terms are cyclic in I, J, K). Writing XH [F ] = {F, H} in coordinates gives I ∂F ∂F ∂H XH I = B JK J K ∂z ∂z ∂z and so ∂H XH = B IJ I . (10.4.3) ∂z J This expression tells us that B IJ should be thought of as the negative inverse of the symplectic matrix, which is literally correct in the nondegenerate case. Indeed, if we write out Ω(XH , v) = dH · v in coordinates, we get ∂H J ∂H ΩIJ XH v J = I v , i.e., I ΩIJ XH = . ∂z J ∂z J If [ΩIJ ] denotes the inverse of [ΩIJ ], we get ∂H XH = ΩJI I , (10.4.4) ∂z J so comparing (10.4.3) and (10.4.4) we see that B IJ = −ΩIJ . ........................... 15 July 1998—18h02 ...........................

358. 344 10. Poisson Manifolds Recalling that the matrix of Ω is the inverse of that of Ω and that the matrix of Ω is the negative of that of Ω, we see that B = Ω . Let us prove this abstractly. The basic link between the Poisson tensor B and the symplectic form Ω is that they give the same Poisson bracket: {F, H} = B(dF, dH) = Ω(XF , XH ), that is, dF, B dH = dF, XH . But Ω(XH , v) = dH · v, and so Ω XH , v = dH, v , whence, XH = Ω dH since Ω = (Ω )−1 . Thus, B dH = Ω dH, for all H, and thus, B =Ω . Coordinate Representation of Poisson Maps. We have seen that the matrix [B IJ ] of the Poisson tensor B converts the diﬀerential ∂H I dH = dz ∂z I of a function to the corresponding Hamiltonian vector ﬁeld; this is consistent with our treatment in the Introduction and Overview. Another basic concept, that of a Poisson map, is also worthwhile working out in coordinates. Let f : P1 → P2 be a Poisson map, so {F ◦ f, G ◦ f }1 = {F, G}2 ◦ f . In coordinates z I on P1 and wK on P2 , and writing wK = wK (z I ) for the map f , this reads ∂ ∂ ∂F ∂G KL (F ◦ f ) J (G ◦ f )B IJ (z) = 1 B (w). ∂z I ∂z ∂wK ∂wL 2 By the chain rule, this is equivalent to ∂F ∂wK ∂G ∂wL IJ ∂F ∂G KL B 1 (z) = B (w). ∂w K ∂z I ∂w L ∂z J ∂wK ∂wL 2 ........................... 15 July 1998—18h02 ...........................

359. 10.4 The Poisson Tensor 345 Since F and G are arbitrary, f is Poisson iff ∂wK ∂wL B IJ (z) 1 = B KL (w). 2 ∂z I ∂z J ∗ ∗ Intrinsically, regarding B1 (z) as a map B1 (z) : Tz P1 × Tz P1 → R, this reads ∗ ∗ B1 (z)(Tz f · αw , Tz f · βw ) = B2 (w)(αw , βw ), (10.4.5) ∗ where αw , βw ∈ Tw P2 and f (z) = w. In analogy with the case of vector fields we shall say that if (10.4.5) holds, then B1 and B2 are f -related and denote it by B1 ∼f B2 In other words, f is Poisson iff B1 ∼f B2 . (10.4.6) Lie Derivative of the Poisson Tensor. The next Proposition is equivalent to the fact that the flows of Hamiiltonian vector fields are Poisson maps. Proposition 10.4.1. For any function H ∈ F(P ), we have £XH B = 0. Proof. By definition, we have B(dF, dG) = {F, G} = XG [F ] for any locally defined functions F and G on P . Therefore, £XH (B(dF, dG)) = £XH {F, G} = {{F, G} , H} . However, since the Lie derivative is a derivation, £XH (B(dF, dG)) = (£XH B)(dF, dG) + B(£XH dF, dG) + B(dF, £XH dG) = (£XH B)(dF, dG) + B(d {F, H} , dG) + B(dF, d {G, H}) = (£XH B)(dF, dG) + {{F, H} , G} + {F, {G, H}} = (£XH B)(dF, dG) + {{F, G} , H} , by the Jacobi identity. It follows that (£XH B)(dF, dG) = 0 for any locally ∗ defined functions F, G ∈ F(U ). Since any element of Tz P can be written as dF (z) for some F ∈ F(U ), U open in P , it follows that £XH B = 0. Pauli–Jost Theorem. Suppose that the Poisson tensor B is strongly nondegenerate, that is, it defines an isomorphism B : dF (z) → XF (z) of ∗ Tz P with Tz P , for all z ∈ P . Then P is symplectic and the symplectic form Ω is defined by the formula Ω(XF , XG ) = {F, G} for any locally defined Hamiltonian vector fields XF and XG . One gets dΩ = 0 from Jacobi’s ........................... 15 July 1998—18h02 ...........................

360. 346 10. Poisson Manifolds identity—see Exercise 5.5-1. This is the Pauli–Jost Theorem, due to Pauli [1953] and Jost [1964]. One may be tempted to formulate the above nondegeneracy assumption in a slightly weaker form involving only the Poisson bracket: suppose that for every open subset V of P , if F ∈ F(V ) and {F, G} = 0 for all G ∈ F(U ) and all open subsets U of V , then dF = 0 on V , that is, F is constant on the connected components of V . This condition does not imply that P is symplectic, as the following counter example shows. Let P = R2 with Poisson bracket. ∂F ∂G ∂F ∂G {F, G} (x, y) = y − . ∂x ∂y ∂y ∂x If {F, G} = 0, for all G, then F must be constant on both the upper and lower half-planes and hence by continuity it must be constant on R2 . However, R2 with this Poisson structure is clearly not symplectic. Characteristic Distribution. The subset B (T ∗ P ) of T P is called the characteristic field or distribution of the Poisson structure; it need not be a subbundle of T P , in general. Note that skew-symmetry of the tensor B is equivalent to (B )∗ = −B , where (B )∗ : T ∗ P → T P is the dual of B . If P is finite dimensional, the rank of the Poisson structure at a point ∗ z ∈ P is defined to be the rank of B (z) : Tz P → Tz P ; in local coordinates, it is the rank of the matrix B IJ (z) . Since the flows of Hamiltonian vector fields preserve the Poisson structure, the rank is constant along such a flow. A Poisson structure for which the rank is everywhere equal to the dimension of the manifold is nondegenerate and hence symplectic. Poisson Immersions and Submanifolds. An injectively immersed submanifold i : S → P is called a Poisson immersion if any Hamil- tonian vector field defined on an open subset of P containing i(S) is in the range of Tz i at all points i(z) for z ∈ S. This is equivalent to the following assertion: Proposition 10.4.2. An immersion i : S → P is Poisson iff it satisfies the following condition. If F, G : V ⊂ S → R, where V is open in S, and if F , G : U → R are extensions of F ◦ i−1 , G ◦ i−1 : i(V ) → R to an open neighborhood U of i(V ) in P , then {F , G}|i(V ) is well defined and independent of the extensions. The immersed submanifold S is thus endowed with an induced Poisson structure and i : S → P becomes a Poisson map. Proof. If i : S → P is an injectively immersed Poisson manifold, then {F , G}(i(z)) = dF (i(z)) · XG (i(z)) = dF (i(z)) · Tz i(v) = d(F ◦ i)(z) · v = dF (z) · v, ........................... 15 July 1998—18h02 ...........................

361. 10.4 The Poisson Tensor 347 where v ∈ Tz S is the unique vector satisfying XG (i(z)) = Tz i(v). Thus, {F , G}(i(z)) is independent of the extension F of F ◦ i−1 . By skew-symmetry of the bracket, it is also independent of the extension G of G ◦ i−1 . Then one can define a Poisson structure on S by setting {F, G} = {F , G}|i(V ) for any open subset V of S. In this way i : S → P becomes a Poisson map since by the computation above we have XG (i(z)) = Tz i(XG ) Conversely, assume that the condition on the bracket stated above holds and let H : U → P be a Hamiltonian defined on an open subset U of P intersecting i(S). Then, by what was already shown, S is a Poisson manifold and i : S → P is a Poisson map. We claim that if z ∈ S is such that i(z) ∈ U , we have XH (i(z)) = Tz i(XH◦i (z)), and thus XH (i(z)) ∈ range Tz i, thereby showing that i : S → P is a Poisson immersion. To see this, let K : U → R be an arbitrary function. We have dK(i(z)) · XH (i(z)) = {K, H}(i(z)) = {K ◦ i, H ◦ i}(z) = d(K ◦ i)(z) · XH◦i (z) = dK(i(z)) · Tz i(XH◦i (z)). Since K is arbitrary, we conclude that XH (i(z)) = Tz i(XH◦i (z)). If S ⊂ P is a submanifold of P and the inclusion i is Poisson, we say that S is a Poisson submanifold of P . Note that the only immersed Poisson submanifolds of a symplectic manifold are those whose range in P is open since for any (weak) symplectic manifold P , we have Tz P = {XH (z) | H ∈ F(U ), U open in P }. Note that any Hamiltonian vector field must be tangent to a Poisson submanifold. Also note that the only Poisson submanifolds of a symplectic manifold P are its open sets. Symplectic Stratifications. Now we come to an important result that states that every Poisson manifold is a union of symplectic manifolds, each of which is a Poisson submanifold. Definition 10.4.3. Let P be a Poisson manifold. We say that z1 , z2 ∈ P are on the same symplectic leaf of P if there is a piecewise smooth curve in P joining z1 and z2 , each segment of which is a trajectory of a locally defined Hamiltonian vector field. This is clearly an equivalence relation and an equivalence class is called a symplectic leaf . The symplectic leaf containing the point z is denoted Σz . ........................... 15 July 1998—18h02 ...........................

362. 348 10. Poisson Manifolds Theorem 10.4.4 (Symplectic Stratification Theorem). Let P be a finite dimensional Poisson manifold. Then P is the disjoint union of its symplectic leaves. Each symplectic leaf in P is an injectively immersed Poisson submanifold and the induced Poisson structure on the leaf is symplectic. The dimension of the leaf through a point z equals the rank of the Poisson structure at that point and the tangent space to the leaf at z equals ∗ B # (z)(Tz P ) = {XH (z) | H ∈ F(U ), U open in P }. The picture one should have in mind is shown in figure 10.4.1. Note in particular that the dimension of the symplectic leaf through a point can change dimension as the point varies. span of the Hamiltonian vector fields XH (z) a two dimensional symplectic leaf Σz P z zero dimensional symplectic leafs (points) Figure 10.4.1. The symplectic leaves of a Poisson manifold. The Poisson bracket on P can be alternatively described as follows. To evaluate the Poisson bracket of F and G at z ∈ P , restrict F and G to the symplectic leaf Σ through z, take their bracket on Σ (in the sense of brackets on a symplectic manifold ), and evaluate at z. Also note that since the Casimir functions have differentials that annihilate the characteristic field, they are constant on symplectic leaves. To get a feeling for the geometric content of the symplectic stratification theorem, let us first prove it under the assumption that the characteristic field is a smooth vector subbundle of T P which is the case considered originally by Lie [1890]. In finite dimensions, this is guaranteed if the rank of the Poisson structure is constant. Jacobi’s identity shows that the characteristic field is involutive and thus by the Frobenius Theorem, it is integrable. ........................... 15 July 1998—18h02 ...........................

363. 10.4 The Poisson Tensor 349 Therefore, P is foliated by injectively immersed submanifolds whose tangent space at any point coincides with the subspace of all Hamiltonian vector fields evaluated at z. Thus, each such leaf Σ is an immersed Poisson submanifold of P . Define the two-form Ω on Σ by Ω(z)(XF (z), XG (z)) = {F, G} (z) for any functions F, G defined on a neighborhood of z in P . Note that Ω is closed by the Jacobi identity (Exercise 5.5-1). Also, if Check x-ref of Exercise. 0 = Ω(z)(XF (z), XG (z)) = dF (z) · XG (z) for all locally defined G, then dF (z)|Tz Σ = d(F ◦ i)(z) = 0 by the Hahn–Banach theorem. Therefore, 0 = XF ◦i (z) = Tz i(XF (z)) = XF (z), since Σ is a Poisson submanifold of P and the inclusion i : Σ → P is a Poisson map, thus showing that Ω is weakly nondegenerate and thereby proving the theorem for the constant rank case. The general case, proved by Kirillov [1976a], is more subtle since for differentiable distributions which are not subbundles, integrability and involutivity are not equivalent. To prove this case, we proceed in a series of technical propositions.5 Proposition 10.4.5. Let P be a finite dimensional Poisson manifold ∗ with Bz : Tz P → Tz P the Poisson tensor. Take z ∈ P and functions f1 , . . . , fk defined on P such that {Bz dfj }1≤j≤k is a basis of the range of Bz . Let Φj,t be the local flow defined in a neighborhood of z generated by the Hamiltonian vector field Xfj = B dfj . Let Ψz1 ,... ,fk (t1 , . . . , tk ) = (Φ1,t1 ◦ · · · ◦ Φk,tk )(z) f for small enough t1 , . . . , tk . Then: (i) There is an open neighborhood Uδ of 0 ∈ Rk such that: Ψz1 ,... ,fk : Uδ → P f is an embedding. (ii) The ranges of (T Ψz1 ,... ,fk )(t) and BΨz f are equal for t ∈ Uδ . f1 ,... ,fk (t) (iii) Ψz1 ,... ,fk (Uδ ) ⊂ Σz . f 5 This proof was kindly supplied by O. Popp ........................... 15 July 1998—18h02 ...........................

364. 350 10. Poisson Manifolds (iv) If Ψy1 ,... ,gk : Uη → P g is another map constructed as above and y ∈ Ψz1 ,... ,fk (Uδ ), then there f is an open subset, U ⊂ Uη , such that Ψy1 ,... ,gk is a diffeomorphism g from U to an open subset in Ψz1 ,... ,fk (Uδ ). f Proof. (i) The smoothness of Ψz1 ,... ,fk follows from the smoothness of f Φj,t in both the flow parameter and manifold variables. Then T0 Ψz1 ,... ,fk (∂/∂tj ) = Xfj (z) = Bz dfj , f which shows that T0 Ψz1 ,... ,fk is injective. It follows that Ψz1 ,... ,fk is an f f embedding on a sufficiently small neighborhood of 0, say Uδ . Notice also that the ranges of T0 Ψz1 ,... ,fk and of Bz coincide. f (ii) From Proposition 10.3.2 we recall that for any invertible Poisson map Φ on P , we have T Φ · Xf = Xf ◦Φ−1 ◦ Φ and from 10.4.1 we know that the Hamiltonian flows are Poisson maps. Therefore, if t = (t1 , . . . , tk ), Tt Ψz1 ,... ,fk (∂/∂tj ) f = (T Φ1,t1 ◦ . . . ◦ T Φj−1,tj−1 ◦ Xfj ◦ Φj+1,tj+1 ◦ . . . ◦ Φk,tk )(z) = (Xhj ◦ Ψz1 ,... ,fk )(t), f where −1 hj = fj ◦ Φ1,t1 ◦ . . . ◦ Φj−1,tj−1 . This shows that range Tt Ψx1 ,... ,fk ⊂ range BΨx f f1 ,... ,fk (t) if t ∈ Uδ . Since B is invariant under Hamiltonian flows, it follows that dim range BΨz = dim range Bz . f1 ,... ,fk (t) This last equality, the previous inclusion, and the last remark in the proof of (i) above conclude (ii). (iii) This is obvious since Ψz1 ,... ,fk is built from piecewise Hamiltonian f curves starting from z. (iv) Note that Xg (z) ∈ range Bz for any z ∈ P and any smooth function g. Using (ii), we see that Xg is tangent to the image of Ψz1 ,... ,fk . f Therefore, the integral curves of Xg remain tangent to Ψz1 ,... ,fk (Uδ ) if f they start from that set. To get Ψy1 ,... ,gk we just have to find Hamiltonian g ........................... 15 July 1998—18h02 ...........................

365. 10.4 The Poisson Tensor 351 curves which start from y. Therefore, we can restrict ourselves to the submanifold Ψz1 ,... ,fk (Uδ ) when computing the flows along the Hamiltonian f vector fields Xgj ; therefore we can consider that the image of Ψy1 ,... ,gk is g in Ψz1 ,... ,fk (Uδ ). The derivative at 0 ∈ Rk of Ψy1 ,... ,gk is an isomorphism f g to the tangent space of Ψz1 ,... ,fk (Uδ ) at y (that is, range By ), using (ii) f above. Thus, the existence of the neighborhood U follows from the inverse function theorem. Proposition 10.4.6. Let P be a Poisson manifold and B its Poisson tensor. Then for each symplectic leaf Σ ⊂ P , the family of charts satisfying (i) in the previous proposition, namely, Ψz1 ,... ,fk | z ∈ Σ, {Bz dfj }1≤j≤k a basis for range Bz f , gives Σ the structure of a differentiable manifold such that the inclusion is an immersion. Then Tz Σ = range Bz (so dim Σ = rank Bz ), for all z ∈ Σ. Moreover, Σ has a unique symplectic structure such that the inclusion is a Poisson map. Proof. Let w ∈ Ψz1 ,... ,fk (Uδ ) ∩ Ψy1 ,... ,gk (U ) and consider Ψw1 ,... ,hk : f g h Uγ → P . Using (iv) in the proposition above, we can choose Uγ small enough so that Ψw1 ,... ,hk (Uγ ) ⊂ Ψz1 ,... ,fk (Uδ ) ∩ Ψy1 ,... ,gk (U ) h f g is a diffeomorphic embedding in both Ψz1 ,... ,fk (Uδ ) and Ψy1 ,... ,gk (U ). This f g shows that the transition maps for the given charts are diffeomorphisms and so define the structure of a differentiable manifold on Σ. The fact that the inclusion is an immersion follows from (i) of the above proposition. We get the tangent space of Σ using (i), (ii) of the previous proposition; then the equality of dimensions follows. It follows from the definition of an immersed Poisson submanifold that Σ is such a submanifold of P . Thus, if i : Σ → P is the inclusion, {f ◦ i, g ◦ i}Σ = {f, g} ◦ i. Hence if {f ◦ i, g ◦ i}Σ (z) = 0 for all functions g then {f, g}(z) = 0 for all g, that is, Xg [f ](z) = 0 for all g. This implies that df |Tz Σ = 0 since the vectors Xg (z) span Tz Σ. Therefore, i∗ df = d(f ◦ i) = 0, which shows that the Poisson tensor on Σ is nondegenerate and thus Σ is a symplectic manifold. This proves the proposition and also completes the proof of the symplectic stratification theorem. Proposition 10.4.7. If P is a Poisson manifold, Σ ⊂ P is a symplectic leaf, and C is a Casimir function, then C is constant on Σ. ........................... 15 July 1998—18h02 ...........................

366. 352 10. Poisson Manifolds Proof. If C were not locally constant on Σ, then there would exist a point z ∈ Σ such that dC(z) · v = 0 for some v ∈ Tz Σ. But Tz Σ is spanned by Xk (z) for k ∈ F(P ) and hence dC(z) · Xk (z) = {C, K}(z) = 0 which implies that dC(z) · v = 0 which is a contradiction. Thus C is locally constant on Σ and hence constant by connectedness of the leaf Σ. There is another proof of the symplectic stratification theorem (using the same idea as for the Darboux coordinates) in Weinstein [1983] (see Libermann and Marle [1987] also.) The proof given above is along the Frobenius integrability idea. Actually it can be used to produce a proof of the generalized Frobenius theorem. Theorem 10.4.8 (Singular Frobenius Theorem). Let D be a distribution of subspaces of the tangent bundle of a finite dimensional manifold M , that is, Dx ⊂ Tx M as x varies in M . Suppose it is smooth in the sense that for each x there are smooth vector fields Xi defined on some open neighborhood of x and with values in D such that Xi (x) give a basis of Dx . Then D is integrable, that is, for each x ∈ M there is an immersed submanifold Σx ⊂ M with Tx Σx = Dx , if and only if the distribution D is invariant under the (local) flows along vector fields with values in D. Proof. The “only if” part follows easily. For the “if” part we remark that the proof of the theorem above can be reproduced here replacing the range of Bz by Dx and the Hamiltonian vector fields with vector fields in D. The crucial property needed to prove (ii) in the above proposition (i.e. Hamiltonian fields remain Hamiltonian under Hamiltonian flows) is replaced by the invariance of D given in the hypothesis. Remarks. 1. The conclusion of the above theorem is the same as the Frobenius integrability theorem but it is not assumed that the dimension of Dx is constant. 2. Analogous to the symplectic leaves of a Poisson manifold, we can define the maximal integral manifolds of the integrable distribution D using curves along vector fields in D instead of Hamiltonian vector fields. They are also injectively immersed submanifolds in M . 3. The condition that (local) flows of the vector fields with values in D leave D invariant implies the involution property of D, that is, [X, Y ] is a vector field with values in D if both X and Y are vector fields with values in D (use (4.3.7)). But the involution property alone is not enough to guarantee that D is integrable (if the dimension of D is not constant). ........................... 15 July 1998—18h02 ...........................

367. 10.4 The Poisson Tensor 353 4. This generalization of the Frobenius integrability theorem is due to Hermann [1964], Stefan [1974], Sussman [1973], and it has proved quite useful in control theory; see also Libermann and Marle [1987]. Examples (a) Let P = R3 with the rigid body bracket. Then the symplectic leaves are spheres centered at the origin. The single point at the origin is the singular leaf in the sense that the Poisson structure has rank zero there. As we shall see later, it is true more generally that the symplectic leaves in g∗ with the Lie–Poisson bracket are the coadjoint orbits. (b) Symplectic leaves need not be submanifolds and one cannot conclude that if all the Casimir functions are constants then the Poisson structure is nondegenerate. For example, consider T3 with a codimension 1 foliation with dense leaves, such as obtained by taking the leaves to be the product of T1 with a leaf of the irrational flow on T2 . Put the usual area element on these leaves and define a Poisson structure on T3 by declaring these to be the symplectic leaves. Any Casimir function is constant, yet the Poisson structure is degenerate. Poisson–Darboux Theorem. Related to the stratification theorem is an analogue of Darboux’ theorem. To state it, first recall from Exercise 10.3-2 that we define the product Poisson structure on P1 × P2 where P1 , P2 are Poisson manifolds by the requirements that the projections π1 : P1 × P2 → ∗ P and π2 : P1 × P2 → P2 are Poisson mappings, and π1 (F(P1 )) and ∗ π2 (F(P2 )) are commuting subalgebras of F(P1 × P2 ). In terms of coordinates, if bracket relations z I , z J = B IJ (z) and wI , wJ = C IJ (w) are given on P1 and P2 , respectively, then these define a bracket on functions of z I and wJ when augmented by the relations z I , wJ = 0. Theorem 10.4.9 (Lie–Weinstein). Let z0 be a point in a Poisson manifold P . There is a neighborhood U of z0 in P and an isomorphism ϕ = ϕS × ϕN : U → S × N , where S is symplectic, N is Poisson, and the rank of N at ϕN (z0 ) is zero. The factors S and N are unique up to local isomorphism. Moreover, if the rank of the Poisson manifold is constant near z0 , there are coordinates (q 1 , . . . , q k , p1 , . . . , pk , y 1 , . . . , y l ) near x0 satisfying the canonical bracket relations q i , q j = {pi , pj } = q i , y j = pi , y j = 0, q i , pj = δj . i When one is proving this theorem, the manifold S can be taken to be the symplectic leaf of P through z0 and N is, locally, any submanifold of P , transverse to S, and such that S ∩ N = {z0 }. In many cases the transverse structure on N is of Lie–Poisson type. For the proof of this theorem and related results, see Weinstein [1983b]; the second part of the theorem is due ........................... 15 July 1998—18h02 ...........................

368. 354 10. Poisson Manifolds to Lie [1890]. For the main examples in this book, we shall not require a detailed local analysis of their Poisson structure, so we shall forego a more detailed study of the local structure of Poisson manifolds. Exercises 10.4-1. If H ∈ F(P ), where P is a Poisson manifold, show that the flow ϕt of XH preserves the symplectic leaves of P . 10.4-2. Let (P, { , }) be a Poisson manifold with Poisson tensor B ∈ Ω2 (P ). Let B : T ∗ P → T P, B (dH) = XH , be the induced bundle map. We shall denote by the same symbol B : Ω1 (P ) → X(P ) the induced map on the sections. The definitions introduced in §10.3 and §10.6 read B(dF, dH) = dF, B (dH) = {F, H} . Define α := B (α). Define for any α, β ∈ Ω1 (P ), {α, β} = −£α β + £β α − d(B(α, β)). (a) Show that if the Poisson bracket on P is induced by a symplectic form Ω, that is, if B = Ω , then B(α, β) = Ω(α , β ). (b) Show that, for any F, G ∈ F(P ), we have {F α, Gβ} = F G {α, β} − F α [G]β + Gβ [F ]α. (c) Show that, for any F, G ∈ F(P ) we have d {F, G} = {dF, dG} . (d) Show that, if α, β ∈ Ω1 (P ) are closed, then, {α, β} = d(B(α, β)). (e) Use £XH B = 0 to show that {α, β} = −[α , β ]. (f) Show that (Ω1 (P ), { , }) is a Lie algebra; that is, prove Jacobi’s identity. 10.4-3 (Weinstein [1983]). Let P be a manifold and X, Y be two linearly independent commuting vector fields. Show that {F, K} = X[F ]Y [K] − Y [F ]X[K] defines a Poisson bracket on P . Show that XH = Y [H]X − X[H]Y. Show that the symplectic leaves are two-dimensional and that their tangent spaces are spanned by X and Y . Show how to get Example (b) preceding 10.4.8 from this construction. ........................... 15 July 1998—18h02 ...........................

369. 10.5 Quotients of Poisson Manifolds 355 10.5 Quotients of Poisson Manifolds Here we shall give the simplest version of a general construction of Poisson manifolds based on symmetry. This construction represents the first steps in a general procedure called reduction. Poisson Reduction Theorem. Suppose that G is a Lie group that acts on a Poisson manifold and that each map Φg : P → P is a Poisson map. Let us also suppose that the action is free and proper, so that the quotient space P/G is a smooth manifold and the projection π : P → P/G is a submersion (see the discussion of this point in §9.3). Theorem 10.5.1. Under these hypotheses, there is a unique Poisson structure on P/G such that π is a Poisson map. See figure 10.5.1. P π orbits of the group action P/G Figure 10.5.1. The quotient of a Poisson manifold by a group action is a Poisson manifold in a natural way. Proof. Let us first assume P/G is Poisson and show uniqueness. The condition that π be Poisson is that for two functions f, k : P/G → R, {f, k} ◦ π = {f ◦ π, k ◦ π}, (10.5.1) where the brackets are on P/G and P , respectively. The function f = f ◦ π is the unique G-invariant function that projects to f . In other words, if [z] ∈ P/G is an equivalence class, whereby g1 ·z and g2 ·z are equivalent, we let f (g · z) = f ([z]) for all g ∈ G. Obviously, this defines f unambiguously, so that f = f ◦ π. We can also characterize this as saying that f assigns ........................... 15 July 1998—18h02 ...........................

370. 356 10. Poisson Manifolds the value f ([z]) to the whole orbit G · z. We can write (10.5.1) as {f, k} ◦ π = {f , k}. Since π is onto, this determines {f, k} uniquely. We can also use (10.5.1) to define {f, k}. First, note that {f , k}(g · z) = {f , k} ◦ Φg (z) = {f ◦ Φg , k ◦ Φg }(z) = {f , k}(z), since Φg is Poisson and since f and k are constant on orbits. Thus, {f , k} is constant on orbits too, and so it defines {f, k} uniquely. It remains to show that {f, k} so defined satisfies the properties of a Poisson structure. However, these all follow from their counterparts on P . For example, if we write Jacobi’s identity on P , namely 0 = {{f , k}, l} + {{l, f }, k} + {{k, l}, f }, it gives, by construction, 0 = {{f, k} ◦ π, l ◦ π} + {{l, f } ◦ π, k ◦ π} + {{k, l} ◦ π, f ◦ π} = {{f, k}, l} ◦ π + {{l, f }, k} ◦ π + {{k, l}, f } ◦ π and thus by surjectivity of π, Jacobi’s identity holds on P/G. This construction is just one of many that produce new Poisson and symplectic manifolds from old ones. We refer to Marsden and Ratiu [1986] and Vaisman [1996] for generalizations of the construction here. Reduction of Dynamics. If H is a G-invariant Hamiltonian on P , it defines a corresponding function h on P/G such that H = h ◦ π. Since π is a Poisson map, it transforms XH on P to Xh on P/G; that is, T π ◦ XH = Xh ◦ π, or XH and Xh are π-related. We say that the Hamiltonian system XH on P reduces to that on P/G. As we shall see in the next chapter, G-invariance of H may be associated with a conserved quantity J : P → R. If it is also G-invariant, the corresponding function j on P/G is conserved for Xh since {h, j} ◦ π = {H, J} = 0 and so {h, j} = 0. Example. Consider the differential equations on C2 given by z1 = −iω1 z1 + i p¯2 + iz1 (s11 |z1 |2 + s12 |z2 |2 ), ˙ z (10.5.2) z2 = −iω2 z2 + i q¯1 − iz2 (s21 |z1 |2 + s22 |z2 |2 ). ˙ z ........................... 15 July 1998—18h02 ...........................

371. 10.5 Quotients of Poisson Manifolds 357 Use the standard Hamiltonian structure obtained by taking the real and imaginary parts of zi as conjugate variables. For example, we write z1 = q1 + ip1 and require q1 = ∂H/∂p1 and p1 = −∂H/∂q1 . Recall from ˙ ˙ Chapter 5 that a useful trick in this regard, that enables one to work in complex notation, is to write Hamilton’s equations as zk = −2i∂H/∂ zk . ˙ ¯ Using this, one readily ﬁnds that (see Exercise 5.4-3): The system (10.5.2) is Hamiltonian if and only if s12 = −s21 and p = q. In this case we can choose H(z1 , z2 ) = 1 (ω2 |z2 |2 + ω1 |z1 |2 ) 2 s11 s12 s22 − p Re(z1 z2 ) − |z1 |4 − |z1 z2 |2 + |z2 |4 . (10.5.3) 4 2 4 Note that for equation (10.5.2) with = 0 there are two copies of S 1 acting on z1 and z2 independently; corresponding conserved quantities are |z1 |2 and |z2 |2 . However, for = 0, the symmetry action is (z1 , z2 ) → (eiθ z1 , e−iθ z2 ) (10.5.4) with the conserved quantity (Exercise 5.5-3) J(z1 , z2 ) = 1 (|z1 |2 − |z2 |2 ). 2 (10.5.5) Let φ = (π/2) − θ1 − θ2 , where z1 = r1 exp(iθ1 ), z2 = r2 exp(iθ2 ). We know that the Hamiltonian structure for (10.5.2) on C2 described above induces one on C2 /S 1 (exclude points where r1 or r2 vanishes), and that the two integrals (energy and the conserved quantity) descend to the quotient space, as does the Poisson bracket. The quotient space C2 /S 1 is parametrized by (r1 , r2 , φ) and H and J can be dropped to the quotient. Concretely, the process of dropping to the quotient is very simple: if F (z1 , z2 ) = F (r1 , θ1 , r2 , θ2 ) is S 1 invariant, then it can be written (uniquely) as a function f of (r1 , r2 , φ). By Theorem 10.5.1, one can also drop the Poisson bracket to the quotient. Consequently, the equations in (r1 , r2 , φ) can be cast in Hamiltonian form f˙ = {f, h} for the induced Poisson bracket. This bracket is obtained by using the chain rule to relate the complex variables and the polar coordinates. One ﬁnds that {f, k}(r1 , r2 , φ) 1 ∂f ∂k ∂f ∂k 1 ∂f ∂k ∂f ∂k =− − − − . (10.5.6) r1 ∂r1 ∂φ ∂φ ∂r1 r2 ∂r2 ∂φ ∂φ ∂r2 The (non-canonical) Poisson bracket (10.5.6) is, of course, the reduction of the original canonical Poisson bracket on the space of q and p variables, written in the new polar coordinate variables. Theorem 10.5.1 shows that Jacobi’s identity is automatic for this reduced bracket. (See Knobloch, Ma- halov, and Marsden [1994] for further examples of this type.) ........................... 15 July 1998—18h02 ...........................

372. 358 10. Poisson Manifolds As we shall see in Chapter 13, a key example of the Poisson reduction given in 10.5.1 is when P = T ∗ G and G acts on itself by left translations. Then P/G ∼ g∗ and the reduced Poisson bracket is none other than the = Lie–Poisson bracket! Exercises 10.5-1. Let R3 be equipped with the rigid body bracket and let G = S 1 act on P = R3 (z-axis) by rotation about the z-axis. Compute the induced bracket on P/G. 10.5-2. Compute explicitly the reduced Hamiltonian h in the example in ˙ ˙ ˙ the text and verify directly that the equations for r1 , r2 , φ are Hamiltonian on C with Hamiltonian h. Also check that the function j induced by J is 2 a constant of the motion. 10.6 The Schouten Bracket The goal of this subsection is to express the Jacobi identity for a Poisson structure in geometric terms analogous to dΩ for symplectic structures. This will be done in terms of a bracket defined on contravariant antisymmetric tensors generalizing the Lie bracket of vector fields (see, for example, Schouten [1940], Nijenhuis [1953], Lichnerowicz [1978], Olver [1984, 1986], Koszul [1985], Libermann and Marle [1987], Bhaskara and Viswanath [1988], Kosman–Schwarzbach and Magri [1990], Vaisman [1994], and references therein). Multivectors. A contravariant antisymmetric q-tensor on a finite- dimensional vector space V is a q-linear map A : V ∗ × V ∗ × · · · × V ∗ (q times) → R that is antisymmetric in each pair of arguments. The space of these tensors will be denoted by Λq (V ). Thus, each element Λq (V ) is a finite linear combination of terms of the form v1 ∧· · ·∧vq , called a q-vector , for v1 , . . . , vq ∈ V . If V is an infinite-dimensional Banach space, we define Λq (V ) to be the span of all elements of the form v1 ∧ · · · ∧ vq with v1 , . . . , vq ∈ V , where the exterior product is defined in the usual manner relative to a weakly nondegenerate pairing , : V ∗ × V → R. Thus, Λ0 (V ) = R and Λ1 (V ) = V . If P is a smooth manifold, let Λq (P ) = Λq (Tz P ), z∈P a smooth vector bundle with fiber over z ∈ P equal to Λq (Tz P ). Let Ωq (P ) denote the smooth sections of Λq (P ), that is, the elements of Ωq (P ) are ........................... 15 July 1998—18h02 ...........................

373. 10.6 The Schouten Bracket 359 smooth contravariant antisymmetric q-tensor fields on P . Let Ω∗ (P ) be the direct sum of the spaces Ωq (P ), where Ω0 (P ) = F(P ). Note that Ωq (P ) = 0 for q dim(P ), and that Ω1 (P ) = X(P ). If X1 , . . . , Xq ∈ X(P ), X1 ∧ · · · ∧ Xq is called a q-vector field , or a mul- tivector field . On the manifold P , consider a (q + p)-form α and a contravariant antisymmetric q-tensor A. The interior product iA α of A with α is defined as follows. If q = 0, so A ∈ R, let iA α = Aα. If q ≥ 1 and if A = v1 ∧· · ·∧vq , where vi ∈ Tz P, i = 1, . . . , q, define iA α ∈ Ωp (P ) by (iA α)(vq+1 , . . . , vq+p ) = α(v1 , . . . , vq+p ) (10.6.1) for arbitrary vq+1 , . . . , vq+p ∈ Tz P . One checks that the definition does not depend on the representation of A as a q-vector, so iA α is well defined on Λq (P ) by linear extension. In local coordinates, for finite-dimensional P , (iA α)iq+1 ...iq+p = Ai1 ...iq αi1 ...iq+p , (10.6.2) where all components are nonstrict. If P is finite dimensional and p = 0, (10.6.1) defines an isomorphism of Ωq (P ) with Ωq (P ). If P is a Banach manifold, (10.6.1) defines a weakly nondegenerate pairing of Ωq (P ) with Ωq (P ). If A ∈ Ωq (P ), q is called the degree of A and is denoted by deg A. One checks that iA∧B α = iB iA α. (10.6.3) The Lie derivative £X is a derivation relative to ∧, that is, £X (A ∧ B) = (£X A) ∧ B + A ∧ (£X B) for any A, B ∈ Ω∗ (P ). The Schouten Bracket. The next theorem produces an interesting bracket on multivectors. Theorem 10.6.1 (Schouten Bracket Theorem). There is a unique bilinear operation [ , ] : Ω∗ (P ) × Ω∗ (P ) → Ω∗ (P ) natural with respect to restriction to open sets, called the Schouten bracket, that satisfies the following properties: (i) it is a biderivation of degree −1, that is, it is bilinear, deg[A, B] = deg A + deg B − 1, (10.6.4) and for A, B, C ∈ Ω∗ (P ), [A, B ∧ C] = [A, B] ∧ C + (−1)(deg A+1) deg B B ∧ [A, C]; (10.6.5) ........................... 15 July 1998—18h02 ...........................

374. 360 10. Poisson Manifolds (ii) it is determined on F(P ) and X(P ) by (a) [F, G] = 0, for all F, G ∈ F(P ); (b) [X, F ] = X[F ], for all F ∈ F(P ), X ∈ X(P ); (c) [X, Y ] for all X, Y ∈ X(P ) is the usual Jacobi–Lie bracket of vector fields; and (iii) [A, B] = (−1)deg A deg B [B, A]. In addition, the Schouten bracket satisfies the graded Jacobi identity (−1)deg A deg C [[A, B], C] + (−1)deg B deg A [[B, C], A] deg C deg B + (−1) [[C, A], B] = 0. (10.6.6) Proof. The proof proceeds in standard fashion and is similar to that characterizing the exterior or Lie derivative by its properties, see (Abra- ham, Marsden, and Ratiu [1988]): on functions and vector fields it is given by (ii); then (i) and linear extension determine it on any skew-symmetric contravariant tensor in the second variable and a function and vector field in the first; (iii) tells how to switch such variables and finally (i) again defines it on any pair of skew-symmetric contravariant tensors. The operation so defined satisfies (i), (ii), and (iii) by construction. Uniqueness is a consequence of the fact that the skew-symmetric contravariant tensors are generated as an exterior algebra locally by functions and vector fields and (ii) gives these. The graded Jacobi identity is verified on an arbitrary triple of q-, p-, and r-vectors using (i), (ii), and (iii) and then invoking trilinearity of the identity. Properties. The following formulas are useful in computing with the Schouten bracket. If X ∈ X(P ) and A ∈ Ωp (P ), induction on the degree of A and the use of property (i) show that [X, A] = £X A. (10.6.7) An immediate consequence of this formula and the graded Jacobi identity is the derivation property of the Lie derivative relative to the Schouten bracket, that is, £X [A, B] = [£X A, B] + [A, £X B], (10.6.8) for A ∈ Ωp (P ), B ∈ Ωq (P ), and X ∈ X(P ). Using induction on the number of vector fields, (10.6.7), and the properties in Theorem 10.6.1, one can prove that r [X1 ∧ · · · ∧ Xr , A] = ˇ (−1)i+1 X1 ∧ · · · ∧ Xi ∧ · · · ∧ Xr ∧ (£Xi A), i=1 (10.6.9) ........................... 15 July 1998—18h02 ...........................

375. 10.6 The Schouten Bracket 361 ˇ where X1 , . . . , Xr ∈ X(P ) and Xi means that Xi has been omitted. The last formula plus linear extension can be taken as the deﬁnition of the Schouten bracket and one can deduce Theorem 10.6.1 from it; see Vaisman [1994] for this approach. If A = Y1 ∧ · · · ∧ Ys for Y1 , · · · , Ys ∈ X(P ), the formula above plus the derivation property of the Lie derivative give [X1 ∧ · · · ∧ Xr , Y1 ∧ · · · ∧ Ys ] r s = (−1)r+1 ˇ (−1)i+j [Xi , Yj ] ∧ X1 ∧ · · · ∧ Xi ∧ · · · i=1 j=1 ˇ ∧ Xr ∧ Y1 ∧ · · · ∧ Yj ∧ · · · ∧ Ys . (10.6.10) Finally, if A ∈ Ωp (P ), B ∈ Ωq (P ), and α ∈ Ωp+q−1 (P ), the formula i[A,B] α = (−1)q(p+1) iA d iB α + (−1)p iB d iA α − iB iA dα (10.6.11) (which is a direct consequence of (10.6.10) and Cartan’s formula for dα) can be taken as the deﬁnition of [A, B] ∈ Ωp+q−1 (P ); this is the approach taken originally in Nijenhuis [1955]. Coordinate Formulas. In local coordinates, denoting ∂/∂z i = ∂i , the formulas (10.6.9) and (10.6.10) imply that 1. for any functon f , p f, ∂i1 ∧ . . . ∧ ∂ip = ˇ (−1)k−1 (∂ik f ) ∂i1 ∧ · · · ∧ ∂ik ∧ · · · ∧ ∂ip k=1 whereˇover a symbol means that it is deleted, and 2. ∂i1 ∧ · · · ∧ ∂ip , ∂j1 ∧ · · · ∧ ∂jq = 0. Therefore, if A = Ai1 ...ip ∂i1 ∧ · · · ∧ ∂ip and B = B j1 ...jq ∂j1 ∧ · · · ∧ ∂jq , we get [A, B] = A i1 ...i −1 i +1 ...ip ∂ B j1 ...jq ∂i1 ∧ · · · ∧ ∂i −1 ∧ ∂i +1 ∧ ∂j1 ∧ · · · ∧ ∂jq + (−1)p B j1 ...j −1 j +1 ...jq ∂ Ai1 ...ip ∂i1 ∧ · · · ∧ ∂ip ∧ ∂j1 ∧ · · · ∧ ∂j −1 ∧ ∂j +1 ∧ · · · ∧ ∂jq (10.6.12) or, more succinctly, k2 ...kp+q k ...k ∂ j1 ...jq i2 ...ip [A, B] = εi22...ipp+q q A j1 ...j B ∂x k ...k ∂ i1 ...iq + (−1)p εi12...ipp+q q B j2 ...jp j2 ...j A (10.6.13) ∂x ........................... 15 July 1998—18h02 ...........................

376. 362 10. Poisson Manifolds where all components are nonstrict. Here i ...i εj1 ...jp+q 1 p+q is the Kronecker symbol : it is zero if (i1 , . . . , ip+q ) = (j1 , . . . , jp+q ), and is 1 (resp., −1) if j1 , . . . , jp+q is an even (resp., odd) permutation of i1 , . . . , ip+q . From §10.6 the Poisson tensor B ∈ Ω2 (P ) deﬁned by a Poisson bracket { , } on P satisﬁes B(dF, dG) = {F, G} for any F, G ∈ F(P ). By (10.6.2), this can be written {F, G} = iB (dF ∧ dG), (10.6.14) or in local coordinates, ∂F ∂G {F, G} = B IJ . ∂z I ∂z J Writing B locally as a sum of terms of the form X ∧ Y for some X, Y ∈ X(P ) and taking Z ∈ X(P ) arbitrarily, by (10.6.1), we have for F, G, H ∈ F(P ), iB (dF ∧ dG ∧ dH)(Z) = (dF ∧ dG ∧ dH)(X, Y, Z)   dF (X) dF (Y ) dF (Z) = det  dG(X) dG(Y ) dG(Z)  dH(X) dH(Y ) dH(Z) dF (X) dF (Y ) dH(X) dH(Y ) = det dH(Z) + det dG(Z) dG(X) dG(Y ) dF (X) dF (Y ) dG(X) dG(Y ) + det dF (Z) dH(X) dH(Y ) = iB (dF ∧ dG)dH(Z) + iB (dH ∧ dF )dG(Z) + iB (dG ∧ dH)dF (Z), that is, iB (dF ∧ dG ∧ dH) = iB (dF ∧ dG)dH + iB (dH ∧ dF )dG + iB (dG ∧ dH)dF. (10.6.15) The Jacobi–Schouten Identity. Equations (10.6.14) and (10.6.15) imply {{F, G} , H} + {{H, F } , G} + {{G, H} , F } = iB (d {F, G} ∧ dH) + iB (d {H, F } ∧ dG) + iB (d {G, H} ∧ dF ) = iB d(iB (dF ∧ dG)dH + iB (dH ∧ dF )dG + iB (dG ∧ dH)dF ) = iB d iB (dF ∧ dG ∧ dH) = 1 i[B,B] (dF ∧ dG ∧ dH), 2 ........................... 15 July 1998—18h02 ...........................

377. 10.6 The Schouten Bracket 363 the last equality being a consequence of (10.6.11). We summarize what we have proved. Theorem 10.6.2. The following identity holds. {{F, G} , H} + {{H, F } , G} + {{G, H} , F } = 1 i[B,B] (dF ∧ dG ∧ dH) 2 (10.6.16) This result shows that Jacobi’s identity for { , } is equivalent to [B, B] = 0. Thus, a Poisson structure is uniquely defined by a contravariant antisymmetric two-tensor whose Schouten bracket with itself vanishes. The local formula (10.6.13) becomes n IJK ∂B IJ ∂B JK ∂B KI [B, B] = B LK + B LI + B LJ ∂z L ∂z L ∂z L L=1 which coincides with our earlier expression (10.4.2). The Lie–Schouten Identity. There is another interesting identity that gives the Lie derivative of the Poisson tensor along a Hamiltonian vector field. Theorem 10.6.3. The following identity holds £XH B = i[B,B] dH. (10.6.17) Proof. In coordinates, ∂B IJ IJ ∂X J ∂X I (£X B) K − B IK K − B KJ K = XK ∂z ∂z ∂z I IJ J so if X = B (∂H/∂z ), this becomes IJ ∂B IJ ∂H ∂ ∂H (£XH B) = B KL − B IK K B JL ∂z K ∂z L ∂z ∂z L ∂ ∂H + B JK K B IL L ∂z ∂z ∂B IJ ∂B JL ∂B IL ∂H = − B IK B KL − B KJ ∂z K ∂z K ∂z K ∂z L LIJ ∂H IJ = [B, B] = i[B,B] dH , ∂z L so (10.6.17) follows. This identity shows how Jacobi’s identity [B, B] = 0 is directly used to show that the flow ϕt of a Hamiltonian vector field is Poisson. The above derivation shows that the flow of a time-dependent Hamiltonian vector field consists of Poisson maps; indeed, even in this case, d (ϕ∗ B) = ϕ∗ (£XH B) = ϕ∗ i[B,B] dH = 0 dt t t t is valid. ........................... 15 July 1998—18h02 ...........................

378. 364 10. Poisson Manifolds Exercises 10.6-1. Prove the following formulas by the method indicated in the text. (a) If A ∈ Ωq (P ) and X ∈ X(P ), then [X, A] = £X A. (b) If A ∈ Ωq (P ) and X1 , . . . , Xr ∈ X(P ), then r [X1 ∧ · · · ∧ Xr , A] = ˇ (−1)i+1 X1 ∧ · · · ∧ Xi ∧ · · · ∧ Xr ∧ (£Xi A). i=1 (c) If X1 , . . . , Xr , Y1 , . . . , Ys ∈ X(P ), then [X1 ∧ · · · ∧ Xr , Y1 ∧ · · · ∧ Ys ] r s = (−1)r+1 (−1)i+j [Xi , Yi ] ∧ i=1 j=1 ˇ ˇ ∧ X1 ∧ · · · ∧ Xi ∧ · · · ∧ Xr ∧ Y1 ∧ · · · ∧ Yj ∧ · · · ∧ Ys . (d) If A ∈ Ωp (P ), B ∈ Ωq (P ), and α ∈ Ωp+q−1 (P ), then i[A,B] α = (−1)q(p+1) iA d iB α + (−1)p iB d iA α − iB iA dα. 10.6-2. Let M be a finite-dimensional manifold. A k-vector field is a ∗ ∗ skew-symmetric contravariant tensor field A(x) : Tx M × · · · × Tx M → R ∗ (k copies of Tx M ). Let x0 ∈ M be such that A(x0 ) = 0. (a) If X ∈ X(M ), show that (£X A)(x0 ) depends only on X(x0 ), thereby defining a map dx0 A : Tx0 M → Tx0 M ∧ · · · ∧ Tx0 M (k times), called the intrinsic derivative of A at x0 . ∗ (b) If α1 , . . . , αk ∈ Tx M, v1 , . . . , vk ∈ Tx M , show that α1 ∧ · · · ∧ αk , v1 ∧ · · · ∧ vk := det [ αi , vj ] ∗ ∗ defines a nondegenerate pairing between Tx M ∧· · ·∧Tx M and Tx M ∧ · · ·∧Tx M . Conclude that these two spaces are dual to each other, that the space Ωk (M ) of k-forms is dual to the space of k-contravariant skew-symmetric tensor fields Ωk (M ), and that the bases dxi1 ∧ · · · ∧ dxik i1 · · · ik and ∂ ∂ i1 ∧ ··· ∧ i i1 · · · ik ∂x ∂x k are dual to each other. ........................... 15 July 1998—18h02 ...........................

379. 10.7 Generalities on Lie–Poisson Structures 365 (c) Show that the dual map (dx0 A)∗ : Tx0 M ∧ · · · ∧ Tx0 M → Tx0 M, ∗ ∗ ∗ is given by (dx0 A)∗ (α1 ∧ · · · ∧ αk ) = d(A(α1 , . . . , αk ))(x0 ), ˜ ˜ where α1 , . . . , αk ∈ Ω1 (M ) are arbitrary one-forms whose values at ˜ ˜ x0 are α1 , . . . , αk . 10.6-3 (Weinstein [1983]). Let (P, { , }) be a finite-dimensional Pois- son manifold with Poisson tensor B ∈ Ω2 (P ). Let z0 ∈ P be such that ∗ B(z0 ) = 0. For α, β ∈ Tz0 P , define [α, β]B = (dz0 B)∗ (α ∧ β) = d(B(˜ , β))(z0 ) α ˜ ˜ ˜ where dz0 B is the intrinsic derivative of B and α, β ∈ Ω1 (P ) are such that ˜ 0 ) = β. (See Exercise 10.6-2.) Show that (α, β) → [α, β]B α(z0 ) = α, β(z ˜ ∗ ∗ ∗ defines a bilinear skew-symmetric map Tz0 P ×Tz0 P → Tz0 P . Show that the Jacobi identity for the Poisson bracket implies that [ , ]B is a Lie bracket on ∗ ∗ Tz0 P . Since (Tz0 P, [ , ]B ) is a Lie algebra, its dual Tz0 P naturally carries the induced Lie–Poisson structure, called the linearization of the given Poisson bracket at z0 . Show that the linearization in local coordinates has the expression ∂B ij (z0 ) ∂F ∂G k {F, G} (v) = v , ∂z k ∂v i ∂v j for F, G : Tz0 P → R and v ∈ Tz0 P . 10.6-4 (Magri–Weinstein). On the finite-dimensional manifold P , assume one has a symplectic form Ω and a Poisson structure B. Denote by K = B ◦ Ω : T P → T P . Show that (Ω )−1 + B : T ∗ P → T P defines a new Poisson structure on P if and only if Ω ◦ K n induces a closed two form (called a presymplectic form) on P for all n ∈ N. 10.7 Generalities on Lie–Poisson Structures The Lie–Poisson Equations. We begin by working out Hamilton’s equations for the Lie–Poisson bracket. Proposition 10.7.1. Let G be a Lie group. The equations of motion for the Hamiltonian H with respect to the ± Lie–Poisson brackets on g∗ are dµ = ad∗ δH/δµ µ. (10.7.1) dt ........................... 15 July 1998—18h02 ...........................

380. 366 10. Poisson Manifolds Proof. Let F ∈ F(g∗ ) be an arbitrary function. By the chain rule, dF δF = DF (µ) · µ = ˙ µ, ˙ , (10.7.2) dt δµ while δF δH δF {F, H}± (µ) = ± µ, , = ± µ, − adδH/δµ δµ δµ δµ δF = ad∗ δH/δµ µ, . (10.7.3) δµ Nondegeneracy of the pairing and arbitrariness of F imply the result. Caution. In infinite dimensions, g∗ does not necessarily mean the literal functional analytic dual of g, but rather a space in (nondegenerate) duality with g. In this case, care must be taken with the definition of δF/δµ. Formula (10.7.1) says that on g∗ , the Hamiltonian vector field of H : ± g∗ → R is given by XH (µ) = ad∗ δH/δµ µ. (10.7.4) For example, for G = SO(3), formula (10.1.3) for the Lie–Poisson bracket gives XH (Π) = Π × H. (10.7.5) Historical Note. Lagrange devoted a good deal of attention in Vol- ume 2 of Mćanique Analytique [1788] to the study of rotational motion e of mechanical systems. In fact, in equation A on page 212 he gives the reduced Lie–Poisson equations for SO(3) for a rather general Lagrangian. This equation is essentially the same as (10.7.5). His derivation was just how we would do it today—by reduction from material to spatial representation. Formula (10.7.5) actually hides a subtle point in that it identifies g and g∗ . Indeed, the way Lagrange wrote the equations, they are much more like their counterpart on g, which are called the Euler–Poincar´ equations. e We will come to these in Chapter 13, where additional historical information may be found. Coordinate Formulas. In finite dimensions, if ξa , a = 1, 2, . . . , l, is a d basis for g, the structure constants Cab are defined by d [ξa , ξb ] = Cab ξd (10.7.6) ........................... 15 July 1998—18h02 ...........................

381. 10.7 Generalities on Lie–Poisson Structures 367 (a sum on “d” is understood). Thus, the Lie–Poisson bracket becomes ∂F ∂K d {F, K}± (µ) = ±µd C , (10.7.7) ∂µa ∂µb ab where µ = µa ξ a , {ξ a } is the basis of g∗ dual to {ξa }, and summation on repeated indices is understood. Taking F and K to be components of µ, (10.7.7) becomes {µa , µb }± = ±Cab µd . d (10.7.8) The equations of motion for a Hamiltonian H likewise become d ∂H µa = ˙ µd Cab . (10.7.9) ∂µb Poisson Maps. In the Lie–Poisson reduction theorem in Chapter 13 we will show that the maps from T ∗ G to g∗ (resp., g∗ ) given by αg → Te Lg ·αg − + ∗ ∗ (resp., αg → Te Rg ·αg ) are Poisson maps. We will show in the next chapter that this is a general property of momentum maps. Here is another class of Poisson maps that will also turn out to be momentum maps. Proposition 10.7.2. Let G and H be Lie groups and let g and h be their Lie algebras. Let α : g → h be a linear map. The map α is a homomorphism of Lie algebras if and only if its dual α∗ : h∗ → g∗ is a (linear) Poisson ± ± map. Proof. Let F, K ∈ F(g∗ ). To compute δ(F ◦ α∗ )/δµ, we let ν = α∗ (µ) and use the deﬁnition of the functional derivative and the chain rule to get δ (F ◦ α∗ ), δµ = D(F ◦ α∗ )(µ) · δµ = DF (α∗ (µ)) · α∗ (δµ) δµ δF δF = α∗ (δµ), = δµ, α · . (10.7.10) δν δν Thus, δ δF (F ◦ α∗ ) = α · . (10.7.11) δµ δν Hence, δ δ {F ◦ α∗ , K ◦ α∗ }+ (µ) = µ, (F ◦ α∗ ), (K ◦ α∗ ) δµ δµ δF δK = µ, α · , α· . (10.7.12) δν δν The expression (10.7.12) equals δF δG µ, α · , (10.7.13) δν δν for all F and K if and only if α is a Lie algebra homomorphism. ........................... 15 July 1998—18h02 ...........................

382. 368 10. Poisson Manifolds This theorem applies to the case α = Te σ for σ : G → H a Lie group homomorphism, by studying the reduction diagram in Figure 10.7.1 (and being cautious that σ need not be a diffeomorphism.) T ∗σ Tg G ' ∗ ∗ Tσ(g) H right right translate to translate to identity c cidentity g∗ ' + h∗ + α∗ Figure 10.7.1. Lie group homomorphisms induce Poisson maps. Examples (a) Plasma to Fluid Poisson Map for the Momentum Variables. Let G be the group of diffeomorphisms of a manifold Q and let H be the group of canonical transformations of P = T ∗ Q. We assume that the topology of Q is such that all locally Hamiltonian vector fields on T ∗ Q are globally Hamiltonian.6 Thus, the Lie algebra h consists of functions on T ∗ Q modulo constants. Its dual is identified with itself via the L2 -inner product relative to the Liouville measure dq dp on T ∗ Q. Let σ : G → H be the map η → T ∗ η −1 , which is a group homomorphism and let α = Te σ : g → h. We claim that α∗ : F(T ∗ Q)/R → g∗ is given by α∗ (F ) = pf (q, p) dp, (10.7.14) where we regard g∗ as the space of one-form densities on Q and the integral denotes fiber integration for each fixed q ∈ Q. Indeed, α is the map taking vector fields X on Q to their lifts XP(X) on T ∗ Q. Thus, as a map of X(Q) to F(T ∗ Q)/R, α is given by X → P(X). Its dual is given by α∗ (f ), X = f, α(X) = f P(X) dq dp P = f (q, p)p · X(q) dq dp (10.7.15) P so α∗ (F ) is given by (10.7.14) as claimed. 6 For example, this holds if the first cohomology group H 1 (Q) is trivial. ........................... 15 July 1998—18h02 ...........................

383. 10.7 Generalities on Lie–Poisson Structures 369 (b) Plasma to Fluid Map for the Density Variable. Let G = F(Q), regarded as an abelian group and let the map σ : G → Diff can (T ∗ Q) be given by σ(ϕ) = fiber translation by dϕ. A computation similar to that above gives the Poisson map α∗ (f )(q) = f (q, p) dp (10.7.16) from F(T ∗ Q) to Den(Q) = F(Q)∗ . The integral in (10.7.16) denotes the fiber integration of f (q, p) for fixed q ∈ Q. Linear Poisson Structures are Lie–Poisson. Next we characterize Lie–Poisson brackets as the linear ones. Let V ∗ and V be Banach spaces and let , : V ∗ × V → R be a weakly nondegenerate pairing of V ∗ with V . Think of elements of V as linear functionals on V ∗ . A Poisson bracket on V ∗ is called linear if the bracket of any two linear functionals on V ∗ is again linear. This condition is equivalent to the associated Poisson tensor B(µ) : V → V ∗ being linear in µ ∈ V ∗ . Proposition 10.7.3. Let , : V ∗ × V → R be a (weakly) nondegenerate pairing of the Banach spaces V ∗ and V , and let V ∗ have a linear Poisson bracket. Assume that the bracket of any two linear functionals on V ∗ is in the range of µ, · for all µ ∈ V ∗ (this condition is automatically satisfied if V is finite dimensional). Then V is a Lie algebra and the Poisson bracket on V ∗ is the corresponding Lie–Poisson bracket. Proof. If x ∈ V , we denote by x the functional x (µ) = µ, x on V ∗ . By hypothesis, the Poisson bracket {x , y } is a linear functional on V ∗ . By assumption this bracket is represented by an element which we denote [x, y] in V , that is, we can write {x , y } = [x, y] . (The element [x, y] is unique since , is weakly nondegenerate.) It is straightforward to check that the operation [ , ] on V so defined is a Lie algebra bracket. Thus, V is a Lie algebra, and one then checks that the given Poisson bracket is the Lie–Poisson bracket for this algebra. Exercises 10.7-1. Let σ : SO(3) → GL(3) be the inclusion map. Identify so(3)∗ = R3 with the rigid body bracket and identify gl(3)∗ with gl(3) using A, B = trace(AB T ). Compute the induced map α∗ : gl(3) → R3 and verify directly that it is Poisson. ........................... 15 July 1998—18h02 ...........................

384. 370 10. Poisson Manifolds ........................... 15 July 1998—18h02 ...........................

385. This is page 371 Printer: Opaque this 11 Momentum Maps In this chapter we show how to obtain conserved quantities for Lagrangian and Hamiltonian systems with symmetries. This is done using the concept of a momentum mapping, which is a geometric generalization of the classical linear and angular momentum. This concept is more than a mathematical reformulation of a concept that simply describes the well-known Noether theorem. Rather, it is a rich concept that is ubiquitous in the modern developments of geometric mechanics. It has led to surprising insights into many areas of mechanics and geometry. 11.1 Canonical Actions and Their Infinitesimal Generators Canonical Actions. Let P be a Poisson manifold, let G be a Lie group, and let Φ : G × P → P be a smooth left action of G on P by canonical transformations. If we denote the action by g · z = Φg (z), so that Φg : P → P , then the action being canonical means Φ∗ {F1 , F2 } = Φ∗ F1 , Φ∗ F2 g g g (11.1.1) for any F1 , F2 ∈ F(P ) and any g ∈ G. If P is a symplectic manifold with symplectic form Ω, then the action is canonical if and only if it is symplectic, that is, Φ∗ Ω = Ω for all g ∈ G. g Infinitessimal Generators. Recall from Chapter 9 on Lie groups, that the infinitesimal generator of the action corresponding to a Lie algebra

386. 372 11. Momentum Maps element ξ ∈ g is the vector field ξP on P obtained by differentiating the action with respect to g at the identity in the direction ξ. By the chain rule, d ξP (z) = [exp(tξ) · z] . (11.1.2) dt t=0 We will need two general identities, both of which were proved in Chapter 9. First, the flow of the vector field ξP is ϕt = Φexp tξ . (11.1.3) Second, we have Φ∗−1 ξP = (Adg ξ)P g (11.1.4) and its differentiated companion [ξP , ηP ] = − [ξ, η]P . (11.1.5) The Rotation Group. To illustrate these identities, consider the action of SO(3) on R3 . As was explained in Chapter 9, the Lie algebra so(3) of SO(3) is identified with R3 and the Lie bracket is identified with the cross product. For the action of SO(3) on R3 given by rotations, the infinitesimal generator of ω ∈ R3 is ωR3 (x) = ω × x = ω (x). ˆ (11.1.6) Then (11.1.4) becomes the identity (Aω × x) = A(ω × A−1 x) (11.1.7) for A ∈ SO(3), while (11.1.5) becomes the Jacobi identity for the vector product. Poisson Automorphisms. Returning to the general case, differentiate (11.1.1) with respect to g in the direction ξ, to give ξP [{F1 , F2 }] = {ξP [F1 ], F2 } + {F1 , ξP [F2 ]} . (11.1.8) In the symplectic case, differentiating Φ∗ Ω = Ω gives g £ξP Ω = 0, (11.1.9) that is, ξP is locally Hamiltonian. For Poisson manifolds, a vector field satisfying (11.1.8) is called an infinitesimal Poisson automorphism. Such a vector field need not be locally Hamiltonian (that is, locally of the form XH ). For example, consider the Poisson structure ∂F ∂H ∂H ∂F {F, H} = x − (11.1.10) ∂x ∂y ∂x ∂y ........................... 15 July 1998—18h02 ...........................

387. 11.2 Momentum Maps 373 on R2 and X = ∂/∂y in a neighborhood of a point of the y-axis. We are interested in the case in which ξP is globally Hamiltonian, a condition stronger than (11.1.8). Thus, assume that there is a global Hamiltonian J(ξ) ∈ F(P ) for ξP , that is, XJ(ξ) = ξP . (11.1.11) Does this equation determine J(ξ)? Obviously not, for if J1 (ξ) and J2 (ξ) both satisfy (11.1.11), then XJ1 (ξ)−J2 (ξ) = 0; i.e., J1 (ξ) − J2 (ξ) ∈ C(P ) the space of Casimir functions on P . If P is symplectic and connected, then J(ξ) is determined by (11.1.11) up to a constant. Exercises 11.1-1. Verify (11.1.4), namely: Φ∗−1 ξP = (Adg ξ)P g and its differentiated companion (11.1.5), namely: [ξP , ηP ] = − [ξ, η]P . for the action of GL(n) on itself by conjugation. 11.1-2. Let S 1 act on S 2 by rotations about the z-axis. Compute J(ξ). 11.2 Momentum Maps Since the right-hand side of (11.1.11) is linear in ξ, by using a basis in the finite dimensional case, we can modify any given J(ξ) so it too is linear in ξ, and still retain condition (11.1.11). Indeed, if e1 , . . . , er is a basis of g, ˜ ˜ let the new momentum map J be defined by J(ξ) = ξ a J(ea ). In the definition of the momentum map, we can replace the left Lie group action by a canonical left Lie algebra action ξ → ξP . In the Poisson manifold context, canonical means that (11.1.8) is satisfied and, in the symplectic manifold context, that (11.1.9) is satisfied. (Recall that for a left Lie algebra action, the map ξ ∈ g → ξP ∈ X(P ) is a Lie algebra antihomomorphism.) Thus, we make the following definition: Definition 11.2.1. Let a Lie algebra g act canonically (on the left) on the Poisson manifold P . Suppose there is a linear map J : g → F(P ) such that XJ(ξ) = ξP (11.2.1) ........................... 15 July 1998—18h02 ...........................

388. 374 11. Momentum Maps for all ξ ∈ g. The map J : P → g∗ defined by J(z), ξ = J(ξ)(z) (11.2.2) for all ξ ∈ g and z ∈ P is called a momentum mapping of the action. Angular Momentum. Consider the angular momentum function for a particle in Euclidean three-space, J(z) = q×p, where z = (q, p). Let ξ ∈ R3 and consider the component of J around the axis ξ, namely, J(z), ξ = ξ·(q×p). One checks that Hamilton’s equations determined by this function of q and p describe infinitesimal rotations about the axis ξ. The defining condition (11.2.1) is a generalization of this elementary statement about angular momentum. Momentum Maps and Poisson Brackets. Recalling that XH [F ] = {F, H}, we see that (11.2.1) can be phrased in terms of the Poisson bracket as follows: for any function F on P and any ξ ∈ g, {F, J(ξ)} = ξP [F ] . (11.2.3) Equation (11.2.2) defines an isomorphism between the space of smooth maps J from P to g∗ and the space of linear maps J from g to F(P ). We think of the collection of functions J(ξ) as ξ varies in g as the components of J. Denote by H(P ) = {XF ∈ X(P ) | F ∈ F(P )} (11.2.4) the Lie algebra of Hamiltonian vector fields on P and by P(P ) = {X ∈ X(P ) | X[{F1 , F2 }] = {X[F1 ], F2 } + {F1 , X[F2 ]}} , (11.2.5) the Lie algebra of infinitesimal Poisson automorphisms of P . By (11.1.8), for any ξ ∈ g we have ξP ∈ P(P ). Therefore, giving a momentum map J is equivalent to specifying a linear map J : g → F(P ) making the diagram in Figure 11.2.1 commute. F → XF E F(P ) P(P ) s d d Jd ξ → ξP d g Figure 11.2.1. The commutative diagram defining a momentum map. ........................... 15 July 1998—18h02 ...........................

389. 11.2 Momentum Maps 375 Since both ξ → ξP and F → XF are Lie algebra antihomomorphisms, for ξ, η ∈ g we get XJ([ξ,η]) = [ξ, η]P = − [ξP , ηP ] = − XJ(ξ) , XJ(η) = X{J(ξ),J(η)} (11.2.6) and so we have the basic identity XJ([ξ,η]) = X{J(ξ),J(η)} . (11.2.7) The preceding development defines momentum maps, but does not tell us how to compute them in examples. We shall concentrate on that aspect in Chapter 12. Building on the above commutative diagram, §11.3 discusses an alternative approach to the definition of the momentum map but it will not be used subsequently in the main text. Rather, we shall give the formulas that will be most important for later applications; the interested reader is referred to Souriau [1970], Weinstein [1977], Abraham and Marsden [1978], Guillemin and Sternberg [1984], and Libermann and Marle [1987] for more information. Some History of the Momentum Map The momentum map can be found in the second volume of Lie [1890], where it appears in the context of homogeneous canonical transformations, in which case its expression is given as the contraction of the canonical one-form with the infinitesimal generator of the action. On page 300 it is shown that the momentum map is canonical and on page 329 that it is equivariant with respect to some linear action whose generators are identified on page 331. On page 338 it is proved that if the momentum map has constant rank (a hypothesis that seems to be implicit in all of Lie’s work in this area), its image is Ad∗ -invariant, and on page 343, actions are classified by Ad∗ -invariant submanifolds. We now present the modern history of the momentum map based on information and references provided to us by B. Kostant and J.-M. Souriau. We would like to thank them for all their help. In Kostant’s 1965 Phillips lectures at Haverford (the notes of which were written by Dale Husemoller), and in the 1965 U.S.-Japan Seminar on Dif- ferential Geometry, Kostant [1966] introduced the momentum map to generalize a theorem of Wang and thereby classified all homogeneous symplectic manifolds; this is called today “Kostant’s coadjoint orbit covering theorem.” These lectures also contained the key points of geometric quantization. Meanwhile, Souriau [1965] introduced the momentum map in his Marseille lecture notes and put it in print in Souriau [1966]. The momentum map finally got its formal definition and its name, based on its physical interpretation, in Souriau [1967a] and its properties of equivariance were studied in Souriau [1967b], where the coadjoint orbit theorem is also formulated. In 1968, the momentum map appeared as a key tool in Kostant ........................... 15 July 1998—18h02 ...........................

390. 376 11. Momentum Maps [1968] and from then on became a standard notion. Souriau [1969] discussed it at length in his book and Kostant [1970] (page 187, Theorem 5.4.1) dealt with it in his quantization lectures. Kostant and Souriau realized its importance for linear representations, a fact apparently not foreseen by Lie (Weinstein [1983a]). Independently, work on the momentum map and the coadjoint orbit covering theorem was done by A. Kirillov. This is described in Kirillov [1976b]. This book was first published in 1972 and states that his work on the classification theorem was done about five years earlier (page 301). The modern formulation of the momentum map was developed in the context of classical mechanics in the work of Smale [1970], who applied it extensively in his topological program for the planar n-body problem. Exercises 11.2-1. Verify that Hamilton’s equations determined by the function J(z), ξ = ξ · (q × p) give the infinitesimal generator of rotations about the ξ-axis. 11.2-2. Verify that J([ξ, η]) = {J(ξ), J(η)} for angular momentum. 11.2-3. (a) Let P be a symplectic manifold and G a Lie group acting canonically on P , with an associated momentum map J : P −→ g∗ . Let S be a symplectic submanifold of P which is invariant under the G-action. Show that the G-action on S admits a mometum map given by J|S . (b) Generalize (a) to the case in which P is a Poisson manifold and S is an immersed G-invariant Poisson submanifold. 11.3 An Algebraic Definition of the Momentum Map This section gives an optional approach to momentum maps and may be skipped on a first reading. The point of departure is the commutative diagram in Figure 11.2.1 plus the observation that the following sequence is exact (that is, the range of each map equals the kernel of the following one): 0 C(P) i F(P) H P(P) π P(P)/H(P) 0 Wendy, this Here, i is the inclusion, π the projection, H(F ) = XF , and H(P ) denotes figure was really hard to the Lie algebra of globally Hamiltonian vector fields on P . Let us investigate edit. Paste ins conditions under which a left Lie algebra action, that is, an antihomomor- from tex do phism ρ : g → P(P ), lifts through H to a linear map J : g → F(P ). not work well. ........................... 15 July 1998—18h02 ...........................

391. 11.3 An Algebraic Definition of the Momentum Map 377 As we have already seen, this is equivalent to J being a momentum map. (The requirement that J be a Lie algebra homomorphism will be discussed later.) If H ◦ J = ρ, then π ◦ ρ = π ◦ H ◦ J = 0. Conversely, if π ◦ ρ = 0, then ρ(g) ⊂ H(P ), so there is a linear map J : g → F(P ) such that H ◦ J = ρ. Thus, the obstruction to the existence of J is π ◦ ρ = 0. If P is symplectic, then P(P ) coincides with the Lie algebra of locally Hamiltonian vector fields and thus P(P )/H(P ) is isomorphic to the first cohomology space H 1 (P ) regarded as an abelian group. Thus, in the symplectic case, π ◦ρ = 0 if and only if the induced mapping ρ : g/ [g, g] → H 1 (P ) vanishes. Here is a list of cases which guarantee that π ◦ ρ = 0: 1. P is symplectic and g/[g, g] = 0. By the first Whitehead lemma, this is the case whenever g is semisimple (see Jacobson [1962] and Guillemin and Sternberg [1984]). 2. P(P )/H(P ) = 0. If P is symplectic this is equivalent to the vanishing of the first cohomology group H 1 (P ). 3. P is exact symplectic, that is, Ω = −dΘ, and Θ is invariant under the g action, that is, £ξP Θ = 0. (11.3.1) This case occurs, for example, when P = T ∗ Q and the action is a lift. In Case 3, there is an explicit formula for the momentum map. Since 0 = £ξP Θ = diξP Θ + iξP dΘ, (11.3.2) it follows that d(iξP Θ) = iξP Ω, (11.3.3) that is, the interior product of ξP with Θ satisfies (11.2.1) and hence the momentum map J : P → g∗ is given by J(z), ξ = (iξP Θ) (z) . (11.3.4) In coordinates, write Θ = pi dq i and define Aja and Baj by ∂ ∂ ξP = ξ a Aja + ξ a Baj . (11.3.5) ∂q j ∂pj Then (11.3.4) reads Ja (q, p) = pi Aia (q, p). (11.3.6) ........................... 15 July 1998—18h02 ...........................

392. 378 11. Momentum Maps The following example shows that ρ does not always vanish. Consider the phase space P = S 1 × S 1 , with the symplectic form Ω = dθ1 ∧ dθ2 , the Lie algebra g = R2 , and the action ∂ ∂ ρ(x1 , x2 ) = x1 + x2 . (11.3.7) ∂θ1 ∂θ2 In this case [g, g] = 0 and ρ : R2 → H 1 (S 1 × S 1 ) is an isomorphism, as can be easily checked. 11.4 Conservation of Momentum Maps One reason that momentum maps are important in mechanics is because they are conserved quantities. Theorem 11.4.1 (Hamiltonian Version of Noether’s Theorem). If the Lie algebra g acts canonically on the Poisson manifold P , admits a momentum mapping J : P → g∗ , and H ∈ F(P ) is g-invariant, that is, ξP [H] = 0 for all ξ ∈ g, then J is a constant of the motion for H, that is, J ◦ ϕt = J, where ϕt is the flow of XH . If the Lie algebra action comes from a canonical left Lie group action Φ, then the invariance hypothesis on H is implied by the invariance condition: H ◦ Φg = H for all g ∈ G. Proof. The condition ξP [H] = 0 implies that the Poisson bracket of J(ξ), the Hamiltonian function for ξP , and H vanishes: {J(ξ), H} = 0. This implies that for each Lie algebra element ξ, J(ξ) is a conserved quantity along the flow of XH . This means that the values of the corresponding g∗ - valued momentum map J are conserved. The last assertion of the theorem follows by differentiating the condition H ◦ Φg = H with respect to g at the identity e in the direction ξ to obtain ξP [H] = 0. We dedicate the rest of this section to a list of concrete examples of momentum maps. Examples (a) The Hamiltonian. On a Poisson manifold P , consider the R-action given by the flow of a complete Hamiltonian vector field XH . A corresponding momentum map J : P → R (where we identify R∗ with R via the usual dot product) equals H. ........................... 15 July 1998—18h02 ...........................

393. 11.4 Conservation of Momentum Maps 379 (b) Linear Momentum. In §6.4 we discussed the N -particle system and constructed the cotangent lift of the R3 -action on R3N (translation on every factor) to be the action on T ∗ R3N ∼ R6N given by = x · (qi , pj ) = (qj + x, pj ), j = 1, . . . , N. (11.4.1) We show that this action has a momentum map and compute it from the definition. In the next chapter, we shall recompute it more easily utilizing further developments of the theory. Let ξ ∈ g = R3 ; the infinitesimal generator ξP at a point (qj , pj ) ∈ R6N = P is given by differentiating (11.4.1) with respect to x in the direction ξ ξP (qj , pj ) = (ξ, ξ, . . . , ξ, 0, 0, . . . , 0). (11.4.2) On the other hand, by definition of the canonical symplectic structure Ω on P , any candidate J(ξ) has a Hamiltonian vector field given by ∂J(ξ) ∂J(ξ) XJ(ξ) (qj , pj ) = ,− . (11.4.3) ∂pj ∂qj Then, XJ(ξ) = ξP implies that ∂J(ξ) ∂J(ξ) =ξ and = 0, 1 ≤ j ≤ N. (11.4.4) ∂pj ∂qj Solving these equations and choosing constants such that J is linear, we get   N N J(ξ)(qj , pj ) =  pj  · ξ, i.e., J(qj , pj ) = pj . (11.4.5) j=1 j=1 This expression is called the total linear momentum of the N -particle system. In this example, Noether’s theorem can be deduced directly as follows. Denote by Jα , qj , pj , the αth components of J, qj and pj , α = α α 1, 2, 3. Given a Hamiltonian H, determining the evolution of the N particle system by Hamilton’s equations, we get   N N N dJα dpj ∂H ∂  = α =− j = − j H. (11.4.6) dt j=1 dt j=1 ∂qα j=1 ∂qα The bracket on the right is an operator that evaluates the variation of the scalar function H under a spatial translation, that is, under the action of the translation group R3 on each of the N coordinate directions. Obviously Jα is conserved if H is translation-invariant, which is exactly the statement of Noether’s theorem. ........................... 15 July 1998—18h02 ...........................

394. 380 11. Momentum Maps (c) Angular Momentum. Let SO(3) act on the configuration space Q = R3 by Φ(A, q) = Aq. We show that the lifted action to P = T ∗ R3 has a momentum map and compute it. First note that if (q, v) ∈ Tq R3 , ∗ then Tq ΦA (q, v) = (Aq, Av). Let A · (q, p) = TAq ΦA−1 (q, p) denote the lift of the SO(3) action to P , and identify covectors with vectors using the ∗ Euclidean inner product. If (q, p) ∈ Tq R3 , then (Aq, v) ∈ TAq R3 , so A · (q, p) , (Aq, v) = (q, p) , A−1 · (Aq, v) = p, A−1 v = Ap, v = (Aq, Ap) , (Aq, v) , that is, A · (q, p) = (Aq, Ap) . (11.4.7) Differentiating with respect to A, we find that the infinitesimal generator corresponding to ξ = ω ∈ so(3) is ˆ ωP (q, p) = (ξq, ξp) = (ω × q, ω × p) . ˆ (11.4.8) As in the previous example, to find the momentum map, we solve ∂J(ξ) ∂J(ξ) = ξq and − = ξp, (11.4.9) ∂p ∂q such that J(ξ) is linear in ξ. A solution is given by J(ξ)(q, p) = (ξq) · p = (ω × q) · p = (q × p) · ω, so that J(q, p) = q × p. (11.4.10) Of course, (11.4.10) is the standard formula for the angular momentum of a particle. In this case, Noether’s theorem states that a Hamiltonian that is rota- tionally invariant has the three components of J as constants of the motion. This example can be generalized as follows. (d) Momentum for Matrix Groups. Let G ⊂ GL(n, R) be a subgroup of the general linear group of Rn . We let G act on Rn by matrix multiplication on the left, that is, ΦA (q) = Aq. As in the previous example, the induced action on P = T ∗ Rn is given by A · (q, p) = (Aq, (AT )−1 p) (11.4.11) and the infinitesimal generator corresponding to ξ ∈ g by ξP (q, p) = (ξq, −ξ T p). (11.4.12) ........................... 15 July 1998—18h02 ...........................

395. 11.4 Conservation of Momentum Maps 381 To find the momentum map, we solve ∂J(ξ) ∂J(ξ) = ξq and = ξ T p, (11.4.13) ∂p ∂q which we can do by choosing J(ξ)(q, p) = (ξq) · p, that is, J(q, p), ξ = (ξq) · p. (11.4.14) If n = 3 and G = SO(3), (11.4.14) is equivalent to (11.4.10). In coordinates, (ξq) · p = ξ ij q j pi , so i [J (q, p)]j = q i pj . If we identify g and g∗ using A, B = tr AB T , then J(q, p) is the projection of the matrix q j pi onto the subspace g. (e) Canonical Momentum on g∗ . Let the Lie group G with Lie algebra g act by the coadjoint action on g∗ endowed with the ± Lie–Poisson structure. Since Adg−1 : g → g is a Lie algebra isomorphism, its dual Ad∗−1 : g∗ → g∗ is a canonical map by Proposition 10.7.2. Let us prove g this fact directly. A computation shows that δF δ F ◦ Ad∗−1 g = Adg , (11.4.15) δ(Ad∗−1 µ) g δµ whence {F, H}± Ad∗−1 µ g δF δH = ± Ad∗−1 µ, , g δ Adg−1 µ δ Ad∗−1 µ ∗ g δ F ◦ Ad∗−1 δ H ◦ Ad∗−1 = ± Ad∗−1 µ, Adg g g g , Adg δµ δµ   ∗ δ F ◦ Ad∗−1 δ H ◦ Ad g−1 = ± µ,   g , δµ δµ = F ◦ Ad∗−1 , H ◦ Ad∗−1 g g ± (µ), that is, the coadjoint action of G on g∗ is canonical. From Proposition 10.7.1, the Hamiltonian vector field for H ∈ F(g∗ ) is given by XH (µ) = ad∗ (δH/δµ) µ. (11.4.16) Since the infinitesimal generator of the coadjoint action corresponding to ξ ∈ g is given by ξg∗ = − ad∗ , it follows that the momentum map of the ξ coadjoint action, if it exists, must satisfy ad∗ ∗ (δJ(ξ)/δµ) µ = − adξ µ (11.4.17) ........................... 15 July 1998—18h02 ...........................

396. 382 11. Momentum Maps for every µ ∈ g∗ , that is, J(ξ)(µ) = ± µ, ξ , which means that J = ± identity on g∗ . (f ) Dual of a Lie Algebra Homomorphism. The plasma to fluid map and averaging over a symmetry group in fluid flows are duals of Lie algebra homomorphisms and provide examples of interesting Poisson maps (see §1.7). Let us now show that all such maps are momentum maps. Let H and G be Lie groups, let A : H → G be a Lie group homomorphism and suppose that α : h → g is the induced Lie algebra homomorphism, so its dual α∗ : g∗ → h∗ is a Poisson map. We assert that α∗ is also a momentum map. Let H act on g∗ by + h · µ = Ad∗ −1 µ A(h) that is, h · µ, ξ = µ, AdA(h)−1 ξ . (11.4.18) Differentiating (11.4.18) with respect to h at e in the direction η ∈ h gives the infinitesimal generator ηg∗ (µ), ξ = − µ, adα(η) ξ = − ad∗ α(η) µ, ξ . (11.4.19) Setting J(µ) = α∗ (µ), that is, J(η)(µ) = J(µ), η = α∗ (µ), η = µ, α(η) , (11.4.20) we get δJ(η) = α(η) δµ and so on g∗ , + XJ(η) (µ) = − ad∗ ∗ δJ(η)/δµ µ = − adα(η) µ = ηg∗ (µ), (11.4.21) so we have proved the assertion. (g) Momentum Maps for Subalgebras. Assume that Jg : P → g∗ is a momentum map of a canonical left Lie algebra action of g on the Poisson manifold P and let h ⊂ g be a subalgebra. Then h also acts canonically on P and this action admits a momentum map Jh : P → h∗ given by Jh (z) = Jg (z)|h. (11.4.22) ........................... 15 July 1998—18h02 ...........................

397. 11.4 Conservation of Momentum Maps 383 Indeed, if η ∈ h, we have ηP = XJg (η) since the g-action admits the momentum map Jg and η ∈ g. Therefore, Jh (η) = Jg (η) for all η ∈ h defines the induced h-momentum map on P . This is equivalent to Jh (z), η = Jg (z), η , for all z ∈ P and η ∈ g which proves formula (11.4.22) . To come. (h) Momentum Map on Projective Space. (i) Momentum Maps for Unitary Representations on Projective Space. Here we discuss the momentum map for an action of a finite dimensional Lie group G on projective space that is induced by a unitary representation on the underlying Hilbert space. Recall from § 5.3 that the unitary group U(H) acts on PH by symplectomorphisms. Due to the difficulties in defining the Lie algebra of U(H) (see the remarks at the end of §9.3) we cannot define the momentum map for the whole unitary group. Let ρ : G → U(H) be a unitary representation of G. We can define the infinitesimal action of its Lie algebra g on PDG , the essential G-smooth part of PH, by d ξPH ([ψ]) = [(exp(tA(ξ)))ψ] = Tψ π(A(ξ)ψ), (11.4.23) dt t=0 where the infinitesimal generator A(ξ) was defined in §9.3, where [ψ] ∈ PDG , and where the projection is denoted π : H{0} → PH. Let ϕ ∈ (Cψ)⊥ and ψ = 1. Since A(ξ)ψ − A(ξ)ψ, ψ ψ ∈ (Cψ)⊥ , we have (iξPH Ω)(Tψ π(ϕ)) = −2 Im A(ξ)ψ − A(ξ)ψ, ψ ψ, ϕ = −2 Im A(ξ)ψ, ϕ . On the other hand, if J : PDG → g∗ is defined by ψ, A(ξ)ψ J([ψ]), ξ = J(ξ)([ψ]) = −i , (11.4.24) ψ 2 then, for ϕ ∈ (Cψ)⊥ and ψ = 1, a short computation gives d d(J(ξ))([ψ])(Tψ π(ϕ)) = J(ξ)([ψ + tϕ]) dt t=0 = −2 Im A(ξ)ψ, ϕ . This shows that the map J defined in (11.4.24) is the momentum map of the G-action on PH. We caution that this momentum map is defined only on a dense subset of the symplectic manifold. Recall that a similar thing happened when we discussed the angular momentum for quantum mechanics in §3.3. ........................... 15 July 1998—18h02 ...........................

398. 384 11. Momentum Maps Exercises 11.4-1. For the action of S 1 on C2 given by eiθ (z1 , z2 ) = (eiθ z1 , e−iθ z2 ), show that the momentum map is J = (|z1 |2 − |z2 |2 )/2. Show that the Hamiltonian given in equation (10.5.3) is invariant under S 1 , so that The- orem 11.4.1 applies. 11.4-2. (Momentum Maps Induced by Subgroups) Consider a Poisson action of a Lie group G on the Poisson manifold P with a momentum map J and let H be a Lie subgroup of G. Denote by i : h → g the inclusion between the corresponding Lie algebras and i∗ : g∗ → h∗ the dual map. Check that the induced H-action on P has a momentum map given by K = i∗ ◦ J, that is, K = J|h. 11.4-3 (Euclidean Group in the Plane). The special Euclidean group SE(2) consists of all transformations of R2 of the form Az + a, where z, a ∈ R2 , and cos θ − sin θ A ∈ SO(2) = matrices of the form . (11.4.25) sin θ cos θ This group is three dimensional, with the composition law (A, a) · (B, b) = (AB, Ab + a) , (11.4.26) identity element (l, 0), and inverse −1 (A, a) = A−1 , −A−1 a . We let SE(2) act on R2 by (A, a) · z = Az + a. Let z = (q, p) denote coordinates on R2 . Since det A = 1, we get Φ∗ (dq ∧ dp) = dq ∧ dp, (A,a) that is, SE(2) acts canonically on the symplectic manifold R2 . Show that this action has a momentum map given by J(q, p) = − 1 (q 2 + p2 ), p, −q . 2 11.5 Equivariance of Momentum Maps Inﬁnitessimal equivariance. Return to the commutative diagram in §11.2 and the relations (11.1.8). Since two of the maps in the diagram are Lie algebra antihomomorphisms, it is natural to ask whether J is a ........................... 15 July 1998—18h02 ...........................

399. 11.5 Equivariance of Momentum Maps 385 Lie algebra homomorphism. Equivalently, since XJ[ξ,η] = X{J(ξ),J(η)} , it follows that J([ξ, η]) − {J(ξ), J(η)} =: Σ(ξ, η) is a Casimir function on P and hence is constant on every symplectic leaf of P . As a function on g × g with values in the vector space C(P ) of Casimir functions on P , Σ is bilinear, antisymmetric, and satisfies Σ(ξ, [η, ζ]) + Σ(η, [ζ, ξ]) + Σ(ζ, [ξ, η]) = 0 (11.5.1) for all ξ, η, ζ ∈ g. One says that Σ is a C(P )-valued 2-cocycle of g; see Souriau [1970] and Guillemin and Sternberg [1984], p. 170, for more information. It is natural to ask when Σ(ξ, η) = 0 for all ξ, η ∈ g. In general, this does not happen and one is led to the study of this invariant. We shall derive an equivalent condition for J : g → F(P ) to be a Lie algebra homomorphism; that is, for Σ = 0, or, in other words, for the following commutation relations to hold: J([ξ, η]) = {J(ξ), J(η)}. (11.5.2) Differentiating relation (11.2.2) with respect to z in the direction vz ∈ Tz P , we get d(J(ξ))(z) · vz = Tz J · vz , ξ (11.5.3) for all z ∈ P, vz ∈ Tz P , and ξ ∈ g. Thus, for ξ, η ∈ g, {J(ξ), J(η)} (z) = XJ(η) [J(ξ)] (z) = d(J(ξ))(z) · XJ(η) (z) = Tz J · XJ(η) (z), ξ = Tz J · ηP (z), ξ . (11.5.4) Note that J([ξ, η])(z) = J(z), [ξ, η] = − J(z), adη ξ = − ad∗ J(z), ξ . η (11.5.5) Consequently, J is a Lie algebra homomorphism if and only if Tz J · ηP (z) = − ad∗ J(z) η (11.5.6) for all η ∈ g, that is, (11.5.2) and (11.5.6) are equivalent. Momentum maps satisfying (11.5.2) (or (11.5.6)) are called infinitesimally equivariant momentum maps and canonical (left) Lie algebra actions admitting infinitesimally equivariant momentum maps are called Hamiltonian actions. With this terminology, we have proved the following proposition: ........................... 15 July 1998—18h02 ...........................

400. 386 11. Momentum Maps Theorem 11.5.1. A canonical left Lie algebra action is Hamiltonian if and only if there is a Lie algebra homomorphism ψ : g → F(P ) such that Xψ(ξ) = ξP for all ξ ∈ g. If ψ exists, an infinitesimally equivariant momentum map J is determined by J = ψ. Conversely, if J is infinitesimally equivariant, we can take ψ = J. Equivariance. Let us justify the terminology “infinitesimally equivariant momentum map.” Suppose the canonical left Lie algebra action of g on P arises from a canonical left Lie group action of G on P , where g is the Lie algebra of G. We say that J is equivariant if the diagram in Figure 11.6.1 commutes, that is, J E P g∗ Φg Ad∗−1 g c c P E g∗ J Figure 11.5.1. Equivariance of momentum maps. Ad∗−1 ◦ J = J ◦ Φg g (11.5.7) for all g ∈ G. Equivalently, equivariance can be reformulated as the identity J(Adg ξ)(g · z) = J(ξ)(z) (11.5.8) for all g ∈ G, ξ ∈ g, and z ∈ P . A (left) canonical Lie group action is called globally Hamiltonian if it has an equivariant momentum map. Differentiating (11.5.7) with respect to g at g = e in the direction η ∈ g shows that equivariance implies infinitesimal equivariance. We shall see shortly that all the preceding examples (except the one in Exercise 11.4- 3) have equivariant momentum maps. Another case of interest occurs in Yang-Mills theory, where the 2-cocycle Σ is related to the anomaly (see Bao and Nair [1985] and references therein). The converse question, “When does infinitesimal equivariance imply equivariance?” is treated in §12.4. Momentum Maps for Compact Groups. In the next chapter we shall see that many momentum maps that occur in exaples are equivariant. The next result shows that for compact groups one can always choose them to be equivariant.1 1 A fairly general context in which non-equivariant momentum maps are unavoidable is given in Marsden, Misiolek, Perlmutter and Ratiu [1998]. ........................... 15 July 1998—18h02 ...........................

401. 11.5 Equivariance of Momentum Maps 387 Theorem 11.5.2. Let G be a compact Lie group acting in a canonical fashion on the Poisson manifold P and having a momentum map J : P → g∗ . Then J can be changed by addition of an element of L(g, C(P )) such that the resulting map is an equivariant momentum map. In particular, if P is symplectic J can be changed by the addition of an element of g∗ on each connected component so that the resulting map is an equivariant momentum map. Proof. For each g ∈ G define Jg (z) = Ad∗−1 J(g −1 · z). g or, equivalently, J g (ξ) = J(Adg−1 ξ) ◦ Φg−1 . Then Jg is also a momentum map for the G-action on P . Indeed, if z ∈ P , ξ ∈ g, and F : P → R we have {F, J g (ξ)}(z) = −dJ g (ξ)(z) · XF (z) = −dJ(Adg−1 ξ)(g −1 · z) · Tz Φg−1 · XF (z) = −dJ(Adg−1 ξ)(g −1 · z) · (Φ∗ XF )(g −1 · z) g = −dJ(Adg−1 ξ)(g −1 · z) · XΦ∗ F (g −1 · z) g = {Φ∗ F, J(Adg−1 ξ)}(g −1 · z) g = (Adg−1 ξ)P [Φ∗ F ](g −1 · z) g = (Φ∗ ξP )[Φ∗ F ](g −1 · z) g g = dF (z) · ξP (z) = {F, J(ξ)}(z). Therefore, {F, J g (ξ) − J(ξ)} = 0 for every F : P → R, that is, J g (ξ) − J(ξ) is a Casimir function on P for every g ∈ G and every ξ ∈ g. Now define J = Jg dg G where dg denotes the Haar measure on G normalized such that the total volume of G is one. Equivalently, this definition states that J (ξ) = J g (ξ) dg G for every ξ ∈ g. By linearity of the Poisson bracket in each factor, it follows that {F, J (ξ)} = {F, J g (ξ)} dg = {F, J(ξ)} dg = {F, J(ξ)}. G G ........................... 15 July 1998—18h02 ...........................

402. 388 11. Momentum Maps Thus J is also a momentum map for the G–action on P and J (ξ) − J(ξ) is a Casimir function on P for every ξ ∈ g, that is, J − J ∈ L(g, C(P )). The momentum map J is equivariant. Indeed, noting that −1 Jg (h · z) = Ad∗ −1 Jh h g (z) and using invariance of the Haar measure on G under translations and inversion, for any h ∈ G, we have after changing variables g = hk in the third equality below, −1 −1 J (h · z) = Ad∗ −1 Jh h g (z) dg = Ad∗ −1 h Jh g (z) dg G G = Ad∗ −1 h Jk (z) dk = Ad∗ −1 J (z). h G Exercises 11.5-1. Show that the map J : S 2 → R given by (x, y, z) → z is a momentum map. 11.5-2. Check directly that angular momentum is an equivariant momentum map, whereas the momentum map in Exercise 11.4-3 is not equivariant. 11.5-3. Prove that the momentum map determined by (11.3.4), namely, J(z), ξ = (iξP Θ) (z) , is equivariant. 11.5-4. Let V (n, k) denote the vector space of complex n × k matrices (n rows, k columns). If A ∈ V (n, k), we denote by A† its adjoint (transpose conjugate). (i) Show that A, B = trace(AB † ) is a Hermitian inner product on V (n, k). (ii) Conclude that V (n, k) as a vector space is symplectic and determine the symplectic form. (iii) Show that the action (U, V ) · A = U AV −1 of U (n) × U (k) on V (n, k) is a canonical action. ........................... 15 July 1998—18h02 ...........................

403. 11.5 Equivariance of Momentum Maps 389 (iv) Compute the inﬁnitesimal generators of this action. (v) Show that J : V (n, k) → u(n)∗ × u(k)∗ given by J(A), (ξ, η) = 1 2 trace(AA† ξ) − 1 2 trace(A† Aη) is the momentum map of this action. Identify u(n)∗ with u(n) by the paring ξ1 , ξ2 = − Re[trace(ξ1 , ξ2 )] = − trace(ξ1 , ξ2 ), and similarly, for u(k)∗ u(k); conclude that J(A) = 1 (−iAA† , A† A) ∈ u(n) × u(k). 2 (vi) Show that J is equivariant. ........................... 15 July 1998—18h02 ...........................

404. 390 11. Momentum Maps ........................... 15 July 1998—18h02 ...........................

405. This is page 391 Printer: Opaque this 12 Computation and Properties of Momentum Maps The previous chapter gave the general theory of momentum maps. In this chapter, we develop techniques for computing them. One of the most important cases is when there is a group action on a cotangent bundle and this action is obtained from lifting an action on the base. These transformations are called extended point transformations. 12.1 Momentum Maps on Cotangent Bundles Momentum Functions. We begin by defining functions on cotangent bundles associated to vector fields on the base. Given a manifold Q, Define the map P : X(Q) → F(T ∗ Q), by P(X)(αq ) = αq , X(q) , ∗ for q ∈ Q and αq ∈ Tq Q, where , denotes the pairing between covectors ∗ α ∈ Tq Q and vectors. We call P(X) the momentum function of X. Definition 12.1.1. Given a manifold Q, let L(T ∗ Q) denote the space of smooth functions F : T ∗ Q → R that are linear on fibers of T ∗ Q. Using coordinates and working in finite dimensions, we can write F, H ∈ L(T ∗ Q) as n n i F (q, p) = X (q)pi , and H(q, p) = Y i (q)pi , i=1 i=1

406. 392 12. Computation and Properties of Momentum Maps for functions X i and Y i . We claim that the standard Poisson bracket {F, H} is again linear on the fibers. Indeed, using summations on repeated indices, ∂F ∂H ∂H ∂F ∂X i j ∂Y i j {F, H}(q, p) = j ∂p − j = p Y k δk − j i pi X k δ k ∂q j ∂q ∂pj ∂q ∂q j and so ∂X i j ∂Y i j {F, H} = Y − X pi . (12.1.1) ∂q j ∂q j Hence L(T ∗ Q) is a Lie subalgebra of F(T ∗ Q). If Q is infinite dimensional, a similar proof, using canonical cotangent bundle charts, works. Lemma 12.1.2 (Momentum Commutator Lemma). The Lie algebras: (i) (X(Q), [ , ]) of vector fields on Q; and (ii) Hamiltonian vector fields XF on T ∗ Q with F ∈ L(T ∗ Q) are isomorphic. Moreover, each of these algebras is anti-isomorphic to (L(T ∗ Q), { , }). In particular, we have {P(X), P(Y )} = −P([X, Y ]). (12.1.2) Proof. Since P(X) : T ∗ Q → R is linear on fibers, it follows that P : X(Q) → L(T ∗ Q). This map is linear and satisfies (12.1.2) since [X, Y ]i = (∂Y i /∂q j ) X j − (∂X i /∂q j ) Y j implies that ∂X i j ∂Y i j −P([X, Y ]) = Y − X pi , ∂q j ∂q j which coincides with {P(X), P(Y )} by (12.1.1). (We leave it to the reader to write out the infinite-dimensional proof.) Furthermore, P(X) = 0 implies that X = 0 by the Hahn–Banach theorem. Finally, (assuming our model space is reflexive) for each F ∈ L(T ∗ Q), define X(F ) ∈ X(Q) by ∗ αq , X(F )(q) = F (αq ), where αq ∈ Tq Q. Then P(X(F )) = F , so P is also surjective, thus proving that (X(Q), [ , ]) and (L(T ∗ Q), { , }) are anti- isomorphic Lie algebras. The map F → XF is a Lie algebra antihomomorphism from the algebra (L(T ∗ Q), { , }) to ({XF | F ∈ L(T ∗ Q)}, [ , ]) by (5.5.6). This map is surjec- check x- tive by definition. Moreover, if XF = 0, then F is constant on T ∗ Q, hence ref.5.5.6; is equal to zero since it is linear on the fibers. this the right one? In quantum mechanics, the Dirac rule associates the differential operator ∂ X= Xj (12.1.3) i ∂q j ........................... 15 July 1998—18h02 ...........................

407. 12.1 Momentum Maps on Cotangent Bundles 393 with the momentum function P(X). (Dirac [1930], §21 and §22.) Thus, if we define PX = P(X), (12.1.2) gives i {PX , PY } = i {P(X), P(Y )} = −i P([X, Y ]) = P[X,Y] . (12.1.4) One can augment (12.1.4) by including lifts of functions on Q. Given f ∈ F(Q), let f ∗ = f ◦ πQ where πQ : T ∗ Q → Q is the projection, so f ∗ is constant on fibers. One finds that {f ∗ , g ∗ } = 0 (12.1.5) and {f ∗ , P(X)} = X[f ]. (12.1.6) Hamiltonian Flows of Momentum Functions. The Hamiltonian flow ϕt of Xf ∗ is fiber translation by −t df , that is, (q, p) → (q, p − tdf (q)). The flow of XP(X) is given by the following: Proposition 12.1.3. If X ∈ X(Q) has flow ϕt , then the flow of XP(X) on T ∗ Q is T ∗ ϕ−t . Proof. If πQ : T ∗ Q → Q denotes the canonical projection, differentiating the relation πQ ◦ T ∗ ϕ−t = ϕt ◦ πQ (12.1.7) at t = 0 gives T πQ ◦ Y = X ◦ πQ , (12.1.8) where d ∗ Y (αq ) = T ϕ−t (αq ) , (12.1.9) dt t=0 so T ∗ ϕ−t is the flow of Y . Since T ∗ ϕ−t preserves the canonical one-form Θ on T ∗ Q, it follows that £Y Θ = 0. Hence iY Ω = −iY dΘ = diY Θ. (12.1.10) By definition of the canonical one-form, iY Θ(αq ) = Θ(αq ), Y (αq ) = αq , T πQ (Y (αq )) = αq , X(q) = P(X)(αq ), (12.1.11) that is, iY Ω = dP(X) so that Y = XP(X) . ........................... 15 July 1998—18h02 ...........................

408. 394 12. Computation and Properties of Momentum Maps Because of this proposition, the Hamiltonian vector field XP(X) on T ∗ Q is called the cotangent lift of X ∈ X(Q) to T ∗ Q. We also use the notation X := XP(X) for the cotangent lift of X. From X{F,H} = −[XF , XH ] and (12.1.2), we get [X , Y ] = [XP(X) , XP(Y ) ] = −X{P(X), P(Y )} = − X−P[X,Y ] = [X, Y ] . (12.1.12) For finite-dimensional Q, in local coordinates, we have n ∂P(X) ∂ ∂P(X) ∂ X : = XP(X) = − i=1 ∂pi ∂q i ∂q i ∂pi ∂ ∂X i ∂ = Xi i − p j i ∂p . (12.1.13) ∂q ∂q j Cotangent Momentum Maps. Perhaps the most important result for the computation of momentum maps is the following. Theorem 12.1.4 (Momentum Maps for Lifted Actions). Suppose that the Lie algebra g acts on the left on the manifold Q, so that g acts on P = T ∗ Q on the left by the canonical action ξP = ξQ , where ξQ is the cotangent lift of ξQ to P and ξ ∈ g. This g-action on P is Hamiltonian with infinitesimally equivariant momentum map J : P → g∗ given by J(αq ), ξ = αq , ξQ (q) = P(ξQ )(αq ). (12.1.14) If g is the Lie algebra of a Lie group G which acts on Q and hence on T ∗ Q by cotangent lift, then J is equivariant. In coordinates q i , pj on T ∗ Q and ξ a on g, (12.1.14) reads Ja ξ a = pi ξQ = pi Aia ξ a , i where ξQ = ξ a Aia are the components of ξQ ; thus, i Ja (q, p) = pi Aia (q). (12.1.15) Proof. For ξ, η ∈ g, (12.1.12) gives [ξ, η]P = [ξ, η]Q = −[ξQ , ηQ ] = −[ξQ , ηQ ] = −[ξP , ηP ] and hence ξ → ξP is a left algebra action. This action is also canonical, for if F, H ∈ F(P ), ξP [{F, H}] = XP(ξQ ) [{F, H}] = XP(ξQ ) [F ], H + F, XP(ξQ ) [H] = {ξP [F ], H} + {F, ξP [H]} ........................... 15 July 1998—18h02 ...........................

409. 12.1 Momentum Maps on Cotangent Bundles 395 by the Jacobi identity for the Poisson bracket. If ϕt is the flow of ξQ , the flow of ξQ = XP(ξQ ) is T ∗ ϕ−t . Consequently, ξP = XP(ξQ ) and, thus, the g-action on P admits a momentum map given by J(ξ) = P (ξQ ). Since ξ ∈ g → P(ξQ ) = J(ξ) ∈ F(P ) is a Lie algebra homomorphism by (11.1.5) and (12.1.12), it follows that J is an infinitesimally equivariant momentum map (Theorem 11.5.1). Equivariance under G is proved directly in the following way. For any g ∈ G, we have J(g · αq ), ξ = g · αq , ξQ (g · q) = αq , (Tg·q Φ−1 ◦ ξQ ◦ Φg )(q) g = αq , (Φ∗ ξQ )(q) g = αq , (Adg−1 ξ)Q (q) (by Lemma 9.3.7ii) = J(αq ), Adg−1 ξ = Ad∗−1 (J(αq )), ξ . g Remarks. 1. Let G = Diff(Q) act on T ∗ Q by cotangent lift. Then the infinitesimal generator of X ∈ X(Q) = g is XP(X) by Proposition 12.1.3 so that the associated momentum map is J : T ∗ Q → X(Q)∗ which is defined through J(X) = P(X) by the above calculations. 2. Momentum Fiber Translations. Let G = F(Q) act on T ∗ Q by fiber translations by df , that is, f · αq = αq + df (q). (12.1.16) Since the infinitesimal generator of ξ ∈ F(Q) = g is the vertical lift of dξ(q) at αq and this in turn equals the Hamiltonian vector field −Xξ◦πQ , we see that the momentum map J : T ∗ Q → F(Q)∗ is given by J(ξ) = −ξ ◦ πQ . (12.1.17) This momentum map is equivariant since πQ is constant on fiber translations. 3. The commutation relations {P(X), P(Y )} = −P([X, Y ]), {P(X), ξ ◦ πQ } = −X[ξ] ◦ πQ , (12.1.18) {ξ ◦ πQ , η ◦ πQ } = 0, ........................... 15 July 1998—18h02 ...........................

410. 396 12. Computation and Properties of Momentum Maps can be rephrased as saying that the pair (J(X), J(f )) fit together to form a momentum map for the semidirect product group Diff(Q) F(Q). This plays an important role in the general theory of semidirect products for which we refer the reader to Marsden, Weinstein, Ratiu, Schmid and Spencer [1983], and Marsden, Ratiu and Weinstein [1984a, b]. The terminology extended point transformations arises for the following reasons. Let Φ : G × Q → Q be a smooth action and consider its lift Φ : G × T ∗ Q → T ∗ Q to the cotangent bundle. The action Φ moves points in the configuration space Q, and Φ is its natural extension to phase space T ∗ Q; in coordinates, the action on configuration points q i → q i induces the following action on momenta: ∂q j p i → pi = pj . (12.1.19) ∂q i Exercises 12.1-1. What is the analogue of (12.1.18), namely {P(X), P(Y )} = −P([X, Y ]), {P(X), ξ ◦ πQ } = −X[ξ] ◦ πQ , {ξ ◦ πQ , η ◦ πQ } = 0, for rotations and translations on R3 ? 12.1-2. Prove (12.1.2), namely {P(X), P(Y )} = −P([X, Y ]). in infinite dimensions. 12.1-3. Prove Theorem 12.1.4 as a consequence of formula (11.3.4), namely, J(z), ξ = (iξP Θ) (z) , and Exercise 11.6-3. 12.2 Examples of Momentum Maps We begin this section with the study of momentum maps on tangent bundles. ........................... 15 July 1998—18h02 ...........................

411. 12.2 Examples of Momentum Maps 397 Proposition 12.2.1. Let the Lie algebra g act on the left on the manifold Q and assume that L : T Q → R is a regular Lagrangian. Endow T Q with the symplectic form ΩL = (FL)∗ Ω, where Ω = −dΘ is the canonical symplectic form on T ∗ Q. Then g acts canonically on P = T Q by d ξP (vq ) = Tq ϕt (vq ), dt t=0 where ϕt is the flow of ξQ and has the infinitesimally equivariant momentum map J : T Q → g∗ given by J(vq ), ξ = FL(vq ), ξQ (q) . (12.2.1) If g is the Lie algebra of a Lie group G and G acts on Q and hence on T Q by tangent lift, then J is equivariant. Proof. Use (11.3.4), a direct calculation or, if L is hyperregular, the following argument. Since FL is a symplectic diffeomorphism, ξ → ξP = (FL)∗ ξT ∗ Q is a canonical left Lie algebra action. Therefore, the composition of FL with the momentum map (12.1.14) is the momentum map of the g- action on T Q. In coordinates (q i , q i ) on T Q and (ξ a ) on g, (12.2.1) reads ˙ ∂L i Ja (q i , q i ) = ˙ A a (q), (12.2.2) ∂ qi ˙ where ξQ (q) = ξ a Aia (q) are the components of ξQ . i Next, we shall give a series of examples of momentum maps. Examples (a) The Hamiltonian. A Hamiltonian H : P → R on a Poisson manifold P having a complete vector field XH is an equivariant momentum map for the R-action given by the flow of XH . (b) Linear Momentum. In the notations of Example (b) of §11.4 we recompute the linear momentum of the N -particle system. Since R3 acts on points (q1 , . . . , qN ) in R3N by x · (qj ) = (qj + x), the infinitesimal generator is ξR3N (qj ) = (q1 , . . . , qN , ξ, . . . , ξ) (12.2.3) (this has the base point (q1 , . . . , qN ) and vector part (ξ, . . . , ξ) (N times)). Consequently, by (12.1.14), an equivariant momentum map J : T ∗ R3N → R3 is given by N N J(ξ)(qj , pj ) = pj · ξ, i.e., J(qj , pj ) = pj . j=1 j=1 ........................... 15 July 1998—18h02 ...........................

412. 398 12. Computation and Properties of Momentum Maps (c) Angular Momentum. In the notation of Example (c) of §11.4, let SO(3) act on R3 by matrix multiplication A · q = Aq. The infinitesimal generator is given by ωR3 (q) = ω q = ω × q where ω ∈ R3 . Consequently, ˆ ˆ by (12.1.14), an equivariant momentum map J : T ∗ R3 → so(3)∗ ∼ R3 is = given by J(q, p), ω = p · ω q = ω · (q × p), ˆ that is, J(q, p) = q × p. (12.2.4) Equivariance in this case reduces to the relation Aq × Ap = A(q × p) for any A ∈ SO(3). If A ∈ O(3) SO(3), such as a reflection, this relation is no longer satisfied; a minus sign appears on the right-hand side, a fact sometimes phrased by stating that angular momentum is a pseudo-vector . On the other hand, letting O(3) act on R3 by matrix multiplication, J is given by the same formula and so is the momentum map of a lifted action and these are always equivariant. We have an apparent contradiction— What is wrong? The answer is that the adjoint action and the isomorphism ˆ : R3 → so(3) are related for the component of −(Identity) in O(3) by AÂ−1 = −(Ax) . Thus, J(q, p) is indeed equivariant as it stands. (One x ˆ does not need a separate terminology like “pseudo-vector” to see what is going on.) (d) Momentum for Matrix Groups. In the notations of Example (d) of §11.4, let the Lie group G ⊂ GL(n, R) act on Rn by A · q = Aq. The infinitesimal generator of this action is given by ξRN (q) = ξq, for ξ ∈ g, the Lie algebra of G, regarded as a subalgebra g ⊂ gl(n, R). By (12.1.14), the lift of the G-action on R3 to T ∗ Rn has an equivariant momentum map J : T ∗ Rn → g∗ given by J(q, p), ξ = p · (ξq) (12.2.5) which coincides with (11.4.14). (e) The Dual of a Lie Algebra Homomorphism. From Example (f) of §11.4 it follows that the dual of a Lie algebra homomorphism α : h → g is an equivariant momentum map which does not arise from an action which is an extended point transformation. Recall that a linear map α : h → g is a Lie algebra homomorphism if and only if the dual map α∗ : g∗ → h∗ is Poisson. (f ) Momentum Maps Induced by Subgroups. If a Lie group action of G on P admits an equivariant momentum map J, and if H is a Lie subgroup of G, then in the notation of Exercise 11.4-2, i∗ ◦ J : P → h∗ is an equivariant momentum map of the induced H-action on P . ........................... 15 July 1998—18h02 ...........................

413. 12.2 Examples of Momentum Maps 399 (g) Products. Let P1 and P2 be Poisson manifolds and let P1 × P2 be the product manifold endowed with the product Poisson structure, that is, if F, G : P1 × P2 → R, then {F, G}(z1 , z2 ) = {Fz2 , Gz2 }1 (z1 ) + {Fz1 , Gz1 }2 (z2 ), where { , }i is the Poisson bracket on Pi , Fz1 : P2 → R is the function obtained by freezing z1 ∈ P1 , and similarly for Fz2 : P1 → R. Let the Lie algebra g act canonically on P1 and P2 with (equivariant) momentum mappings J1 : P1 → g∗ and J2 : P2 → g∗ . Then J = J1 + J2 : P1 × P2 → g∗ , J(z1 , z2 ) = J(z1 ) + J(z2 ) is an (equivariant) momentum mapping of the canonical g-action on the product P1 × P2 . There is an obvious generalization to the product of N Poisson manifolds. Note that Example (b) is a special case of this, for G = R3 for all factors in the product manifold equal to T ∗ R3 . (h) Cotangent Lift on T ∗ G. The momentum map for the cotangent lift of the left translation action of G on G is, by (12.1.14), equal to ∗ JL (αg ), ξ = αg , ξG (g) = αg , Te Rg (ξ) = Te Rg (αg ), ξ , that is, ∗ JL (αg ) = Te Rg (αg ). (12.2.6) Similarly, the momentum map for the lift to T ∗ G of right translation of G on G equals ∗ JR (αg ) = Te Lg (αg ). (12.2.7) Notice that JL is right invariant, whereas JR is left invariant. Both are equivariant momentum maps (JR with respect to Ad∗ , which is a right g action), so they are Poisson maps. The diagram in Figure 12.3.1 summarizes the situation. T ∗G JL = right translation of e JR = left translation of e g∗ + g∗ − Figure 12.2.1. Momentum maps for left and right translations. This diagram is an example of what is called a dual pair ; these illumi- nate the relation between the body and spatial description of rigid bodies and ﬂuids; see Chapter 15 for more information. ........................... 15 July 1998—18h02 ...........................

414. 400 12. Computation and Properties of Momentum Maps (i) Momentum Translation on Functions. Let P = F(T ∗ Q)∗ with the Lie–Poisson bracket given in Example (e) of §10.1. Using the Liouville measure on T ∗ Q and assuming that elements of F(T ∗ Q) fall off rapidly enough at infinity, we identify F(T ∗ Q)∗ with F(T ∗ Q) using the L2 -pairing. Let G = F(Q) (with the group operation being addition) act on P by (ϕ · f )(αq ) = f (αq + dϕ(q)), (12.2.8) that is, in coordinates, ∂ϕ f (q i , pj ) → f q i , pj + . ∂q i The infinitesimal generator is ξP (f )(αq ) = Ff (αq ) · dξ(q), (12.2.9) where Ff is the fiber derivative of f . In coordinates, (12.2.9) reads ∂f ∂ξ ξP (f )(q i , pj ) = · . ∂pj ∂q j Since G is a vector space group, its Lie algebra is also F(Q) and we identify F(Q)∗ with one-form densities on Q. If f, g, h ∈ F(T ∗ Q) we have by Corollary 5.5.7 check ref.5.5.7 f {g, h} dq dp = g{h, f } dq dp. (12.2.10) T ∗Q T ∗Q Next, note that if F, H : P = F(T ∗ Q) → R , then we get by (12.2.10) δF δH XH [F ](f ) = {F, H}(f ) = f , dq dp T ∗Q δf δf δF δH = ,f dq dp. (12.2.11) T ∗Q δf δf On the other hand, by (12.2.9), we have δF ξP [F ](f ) = (Ff · (dξ ◦ πQ )) dq dp, (12.2.12) T ∗Q δf which suggests that the definition of J should be J(f ), ξ = f (αq )ξ(q) dq dp. (12.2.13) T ∗Q Indeed, by (12.2.13), we have δJ(ξ)/δf = ξ ◦ πQ so that δJ(ξ) ,f = {ξ ◦ πQ , f } = Ff · (dξ ◦ πQ ) δf ........................... 15 July 1998—18h02 ...........................

415. 12.2 Examples of Momentum Maps 401 and hence by (12.2.11) δF δJ(ξ) XJ(ξ) [F ](f ) = , f dq dp T ∗Q δf δf δF = (Ff · (dξ ◦ πQ )) dq dp, T ∗Q δf which coincides with (12.2.12) thereby proving that J given by (12.2.13) is the momentum map. In other words, the fiber integral J(f ) = f (q, p) dp, (12.2.14) T ∗Q thought of as a one-form density on Q via (12.2.13), is the momentum map in this case. This momentum map is infinitesimally equivariant. Indeed, if ξ, η ∈ F(Q), we have for f ∈ P , δJ(ξ) δJ(η) {J(ξ), J(η)}(f ) = f , dq dp T ∗Q δf δf = f {ξ ◦ πQ , η ◦ πQ } dq dp T ∗Q = 0 = J([ξ, η])(f ). (j) More Momentum Translations. Let Diff can (T ∗ Q) be the group of symplectic diffeomorphisms of T ∗ Q and, as above, let G = F(Q) act on T ∗ Q by translation with df along the fiber, that is, f · αq = αq + df (q). Since the action of the additive group F(Q) is Hamiltonian, F(Q) acts on Diff can (T ∗ Q) by composition on the right with translations, that is, the action is (f, ϕ) ∈ F(Q) × Diff can (T ∗ Q) → ϕ ◦ ρf ∈ Diff can (T ∗ Q) , where ρf (αq ) = αq + df (q). The infinitesimal generator of this action is given by (see the comment preceding (12.1.17)): ξDiff can (T ∗ Q) (ϕ) = −T ϕ ◦ Xξ◦πQ (12.2.15) for ξ ∈ F(Q) = g, so that the equivariant momentum map of the lifted action J : T ∗ (Diff can (T ∗ Q)) → F(Q)∗ given by (12.1.14) is in this case J(ξ)(αϕ ) = − αϕ , T ϕ ◦ Xξ◦πQ , (12.2.16) where the pairing on the right is between vector fields and one-form densities αϕ . (k) The Divergence of the Electric Field. Let A be the space of vector potentials A on R3 and P = T ∗ A, whose elements are denoted (A, −E) ........................... 15 July 1998—18h02 ...........................

416. 402 12. Computation and Properties of Momentum Maps with A and E vector fields. Let G = F(R3 ) act on A by ϕ · A = A + ϕ. Thus, the infinitesimal generator is ξA (A) = ξ. Hence the momentum map is J(A, −E), ξ = −E · ξ d3 x = (div E)ξ d3 x (12.2.17) (assuming fast enough falloff to justify integration by parts). Thus, J(A, −E) = div E (12.2.18) is the equivariant momentum map. (l) Virtual Work. We usually think of covectors as momenta conjugate to configuration variables. However, covectors can also be thought of as ∗ forces. Indeed, if αq ∈ Tq Q and wq ∈ Tq Q, we think of αq , wq = force × infinitesimal displacement as the virtual work . We now give an example of a momentum map in this context. Consider a region B ⊂ R3 with boundary ∂B. Let C be the space of maps ∗ ϕ : B → R3 . Regard Tϕ C as the space of loads; that is, pairs of maps b : B → R , τ : ∂B → R paired with a tangent vector V ∈ Tϕ C by 3 3 (b, τ ), V = b · V d3 x + τ · V dA. B ∂B Let A ∈ GL(3, R) act on C by ϕ → A ◦ ϕ. The infinitesimal generator of this action is ξC (ϕ)(X) = ξϕ(X) for ξ ∈ gl(3) and X ∈ B. Pair gl(3, R) with itself via A, B = 1 tr (AB). The induced momentum map J : T ∗ C → 2 gl(3, R) is given by J(ϕ, (b, τ )) = ϕ ⊗ b d3 x + ϕ ⊗ τ dA. (12.2.19) B ∂B (This is the “astatic load,” a concept from elasticity; see, for example, Marsden and Hughes [1983].) If we take SO(3) rather than GL(3, R), we get the angular momentum. (m) Momentum Maps for Unitary Representations on Projective Space. Here we show that the momentum map discussed in Example (i) of §11.4 is equivariant. Recall from the discussion at the end of §9.3 that associated to a unitary representation ρ of a Lie group G on a complex Hilbert space ........................... 15 July 1998—18h02 ...........................

417. 12.2 Examples of Momentum Maps 403 H, there are skew adjoint operators A(ξ) for each ξ ∈ g depending linearly on ξ and such that ρ(exp(tξ)) = exp(tA(ξ)). Thus, taking the t-derivative in the formula ρ(g)ρ(exp(tξ))ρ(g −1 ) = exp(tρ(g)A(ξ)ρ(g)−1 ), we get A(Adg ξ) = ρ(g)A(ξ)ρ(g)−1 . (12.2.20) Using formula (11.4.24), namely ψ, A(ξ)ψ J([ψ]), ξ = J(ξ)([ψ]) = −i , (12.2.21) ψ 2 we get ψ, ρ(g)A(ξ)ρ(g)−1 ψ J(Adg ξ)([ψ]) = −i ψ 2 = J(ξ)([ρ(g)−1 ψ]) = J(ξ)(g −1 · [ψ]), which shows that J : PH → g∗ is equivariant. Exercises 12.2-1. Derive the conservation of J given by J(vq ), ξ = FL(vq ), ξQ (q) directly from Hamilton’s variational principle. (This is the way Noether originally derived conserved quantities). 12.2-2. If L is independent of one of the coordinates q i , then it is clear that pi = ∂L/∂ q i is a constant of the motion from the Euler–Lagrange ˙ equations. Derive this from Proposition 12.2.1. 12.2-3. Compute JL and JR for G = SO(3). 12.2-4. Compute the momentum maps determined by spatial translations and rotations for Maxwell’s equations. 12.2-5. Repeat Exercise 12.2-4 for elasticity (the context of Example (l)). 12.2-6. Let P be a symplectic manifold and J : P → g∗ be an (equivariant) momentum map for the symplectic action of a group G on P . Let F be the space of (smooth) functions on P identiﬁed with its dual via integration and equipped with the Lie–Poisson bracket. Let J : F → g∗ be deﬁned by J (f ), ξ) = f J, ξ dµ, ........................... 15 July 1998—18h02 ...........................

418. 404 12. Computation and Properties of Momentum Maps where µ is Liouville measure. Show that J is an (equivariant) momentum map. 12.2-7. (i) Let G act on itself by conjugation. Compute the momentum map of its cotangent lift. (ii) Let N ⊂ G be a normal subgroup so that G acts on N by conjugation. Again, compute the momentum map of the cotangent lift of this conjugation action. 12.3 Equivariance and Infinitesimal Equivariance This optional section explores the equivariance of momentum maps a little deeper. We have just seen that equivariance implies infinitesimal equivariance. In this section, we prove, amongst other things, the converse if G is connected. A Family of Casimir Functions. Introduce the map Γη : G × P → R defined by Γη (g, z) = J(Φg (z)), η − Ad∗−1 J(z), η g for η ∈ g. (12.3.1) Since Γη,g (z) := Γη (g, z) = Φ∗ J(η) (z) − J Adg−1 η (z), g (12.3.2) we get XΓη,g = XΦ∗ J(η) − XJ (Ad η) g g −1 = Φ∗ XJ(η) g − Adg−1 η P = Φ∗ ηP − g Adg−1 η P =0 (12.3.3) by (11.1.4). Therefore, Γη,g is a Casimir function on P , and so is constant on every symplectic leaf of P . Since η → Γη (g, z) is linear for every g ∈ G and z ∈ P , we can define the map σ : G → L(g, C(P )), from G to the vector space of all linear maps of g into the space of Casimir functions C(P ) on P , by σ(g) · η = Γη,g . The behavior of σ under group multiplication is the ........................... 15 July 1998—18h02 ...........................

419. 12.3 Equivariance and Infinitesimal Equivariance 405 following. For ξ ∈ g, z ∈ P , and g, h ∈ G, we have (σ(gh) · ξ) (z) = Γξ (gh, z) = (J (Φgh (z)) , ξ − Ad∗ −1 J(z), ξ (gh) = J (Φg (Φh (z))) , ξ − Ad∗−1 J((Φh (z)) , ξ g + J (Φh (z)) , Adg−1 ξ − Ad∗ −1 J(z), Adg−1 ξ h = Γξ (g, Φh (z)) + ΓAdg−1 ξ (h, z) = (σ(g) · ξ) (Φh (z)) + σ(h) · Adg−1 ξ (z). (12.3.4) Connected Lie group actions admitting momentum maps preserve symplectic leaves. This is because G is generated by a neighborhood of the identity in which each element has the form exp tξ; since (t, z) → (exp tξ) · z is a Hamiltonian flow, it follows that z and Φh (z) are on the same leaf. Thus, (σ(g) · ξ) (z) = (σ(g) · ξ) (Φh (z)) because Casimir functions are constant on leaves. Therefore, σ(gh) = σ(g) + Ad† −1 σ(h), g (12.3.5) where Ad† denotes the action of G on L(g, C(P )) induced via the adjoint g action by (Ad† λ)(ξ) = λ(Adg−1 ξ) g (12.3.6) for g ∈ G, ξ ∈ g, and λ ∈ L(g, C(P )). Cocycles. Mappings σ : G → L(g, C(P )), behaving under group multiplication as in (12.3.5), are called L(g, C(P ))-valued one-cocycles of the group G. A one-cocycle σ is called a one-coboundary if there is a λ ∈ L(g, C(P )) such that σ(g) = λ − Ad† −1 λ g for all g ∈ G. (12.3.7) The quotient space of one-cocycles modulo one-coboundaries is called the first L(g, C(P ))-valued group cohomology of G and is denoted by H 1 (G, L (g, C(P ))); its elements are denoted by [σ], for σ a one-cocycle. At the Lie algebra level, bilinear skew-symmetric maps Σ : g × g → C(P ) satisfying the Jacobi type identity (11.6.1) are called C(P )-valued two- cocycles of g. A cocycle Σ is called a coboundary if there is a λ ∈ L(g, C(P )) such that Σ(ξ, η) = λ([ξ, η]) for all ξ, η ∈ g. (12.3.8) The quotient space of two-cocycles by two-coboundaries is called the second cohomology of g with values in C(P ). It is denoted by H 2 (g, C(P )) and its elements by [Σ]. With these notations we have proved the first two parts of the following proposition: ........................... 15 July 1998—18h02 ...........................

420. 406 12. Computation and Properties of Momentum Maps Proposition 12.3.1. Let the connected Lie group G act canonically on the Poisson manifold P and have a momentum map J. For g ∈ G and ξ ∈ g, define Γξ,g : P → R, Γξ,g (z) = J (Φg (z)) , ξ − Ad∗−1 J(z), ξ . g (12.3.9) Then (i) Γξ,g is a Casimir on P for every ξ ∈ g and g ∈ G. (ii) Defining σ : G → L(g, C(P )) by σ(g) · ξ = Γξ,g , we have the identity σ(gh) = σ(g) + Ad† −1 σ(h). g (12.3.10) (iii) Defining ση : G → C(P ) by ση (g) := σ(g) · η for η ∈ g, we have Te ση (ξ) = Σ(ξ, η) := J([ξ, η]) − {J(ξ), J(η)} . (12.3.11) If [σ] = 0, then [Σ] = 0. (iv) If J1 and J2 are two momentum mappings of the same action with cocycles σ1 and σ2 , then [σ1 ] = [σ2 ]. Proof. Since ση (g)(z) = J(η)(g·z)−J(Adg−1 η)(z), taking the derivative at g = e, we get Te ση (ξ)(z) = dJ(η)(ξP (z)) + J([ξ, η])(z) = XJ(ξ) [J(η)](z) + J([ξ, η])(z) = − {J(ξ), J(η)} (z) + J([ξ, η])(z). (12.3.12) This proves (12.3.11). The second statement in (iii) is a consequence of the definition. To prove (iv) we note that σ1 (g)(z) − σ2 (g)(z) = J1 (g · z) − J2 (g · z) − Ad∗−1 (J1 (z) − J2 (z)). g (12.3.13) However, J1 and J2 are momentum mappings of the same action and, therefore, J1 (ξ) and J2 (ξ) generate the same Hamiltonian vector field for all ξ ∈ g, so J1 − J2 is constant as an element of L(g, C(P )). Calling this element λ, we have σ1 (g) − σ2 (g) = λ − Ad† −1 λ, g (12.3.14) so σ1 − σ2 is a coboundary. Remarks. ........................... 15 July 1998—18h02 ...........................

421. 12.3 Equivariance and Infinitesimal Equivariance 407 1. Part (iv) of this proposition also holds for Lie algebra actions admitting momentum maps with all σ’s replaced by Σ’s; indeed, {J1 (ξ), J1 (η)} = {J2 (ξ), J2 (η)} because J1 (ξ) − J2 (ξ) and J1 (η) − J2 (η) are Casimir functions. 2. If [Σ] = 0, the momentum map J : P → g∗ of the canonical Lie algebra action of g on P can be always chosen to be infinitesimally equivariant, a result due to Souriau [1970] for the symplectic case. To see this, note first that momentum maps are determined only up to elements of L(g, C(P )). Therefore, if λ ∈ L(g, C(P )) denotes the element determined by the condition [Σ] = 0, then J + λ is an infinitesimally equivariant momentum map. 3. The cohomology class [Σ] depends only on the Lie algebra action ρ : g → X(P ) and not on the momentum map. Indeed, because J is determined only up to the addition of a linear map λ : g → C(P ) and denoting Σλ (ξ, η) := (J + λ)([ξ, η]) − {(J + λ)(ξ), (J + λ)(η)} , (12.3.15) we obtain Σλ (ξ, η) = J([ξ, η]) + λ([ξ, η]) − {J(ξ), J(η)} = Σ(ξ, η) + λ([ξ, η]), (12.3.16) that is, [Σλ ] = [Σ]. Letting ρ ∈ H 2 (g, C(P )) denote this cohomology class, J is infinitesimally equivariant if and only if ρ vanishes. There are some cases in which one can predict that ρ is zero: (a) Assume P is symplectic and connected (so C(P ) = R) and suppose that H 2 (g, R) = 0. By the second Whitehead lemma (see Jacobson [1962] or Guillemin and Sternberg [1984]), this is the case whenever g is semisimple; thus semisimple, symplectic Lie algebra actions on symplectic manifolds are Hamiltonian. (b) Suppose P is exact symplectic, −dΘ = Ω, and £ξP Θ = 0. (12.3.17) The proof of equivariance in this case is the following. Assume first that the Lie algebra g has an underlying Lie group G which leaves θ invariant. Since Adg−1 ξ P = Φ∗ ξP , we get from (11.3.4) g J(ξ)(g · z) = (iξP Θ) (g · z) = i(Ad −1 ξ) Θ (z) g P = J Adg−1 ξ (z). (12.3.18) ........................... 15 July 1998—18h02 ...........................

422. 408 12. Computation and Properties of Momentum Maps The proof without the assumption of the existence of the group G is obtained by differentiating the above string of equalities with respect to g at g = e. A simple example in which ρ = 0 is provided by phase-space translations on R2 defined by g = R2 = {(a, b)} , P = R2 = {(q, p)}, and ∂ ∂ (a, b)P = a +b . (12.3.19) ∂q ∂p This action has a momentum map given by J(q, p), (a, b) = ap − bq and Σ ((a1 , b1 ) , (a2 , b2 )) = J ([(a1 , b1 ) , (a2 , b2 )]) − {J (a1 , b1 ) , J (a2 , b2 )} = − {a1 p − b1 q, a2 p − b2 q} = b1 a2 − a1 b2 . (12.3.20) Since [g, g] = {0}, the only coboundary is zero, so ρ = 0. This example is amplified in Example (b) of §12.4. 4. If P is symplectic and connected and σ is a one-cocycle of the G-action on P , then: (a) g · µ = Ad∗−1 µ + σ(g) is an action of G on g∗ ; and g (b) J is equivariant with respect to this action. Indeed, since P is symplectic and connected, C(P ) = R, and thus σ : G → g∗ . By Proposition 12.3.1, (gh) · µ = Ad∗ −1 µ + σ(gh) (gh) = Ad∗−1 Ad∗ −1 µ + σ(g) + Ad∗−1 σ(h) g h g = Ad∗−1 (h · µ) + σ(g) = g · (h · µ), g (12.3.21) which proves (a); (b) is a consequence of the definition. 5. If P is symplectic and connected, J : P → g∗ is a momentum map, and Σ is the associated real-valued Lie algebra two-cocycle, then the momentum map J can be explicitly adjusted to be infinitesimally equivariant by enlarging g to the central extension defined by Σ. Indeed, the central extension defined by Σ is the Lie algebra g := g ⊕ R with the bracket given by [(ξ, a) , (η, b)] = ([ξ, η] , Σ (ξ, η)) . (12.3.22) ........................... 15 July 1998—18h02 ...........................

423. 12.3 Equivariance and Infinitesimal Equivariance 409 Let g act on P by ρ(ξ, a)(z) = ξP (z) and let J : P → (g )∗ = g∗ ⊕ R be the induced momentum map, that is, it satisfies XJ (ξ,a) = (ξ, a)P = XJ(ξ) , (12.3.23) so that J (ξ, a) − J(ξ) = (ξ, a), (12.3.24) where (ξ, a) is a constant on P and is linear in (ξ, a). Therefore, J ([(ξ, a) , (η, b)]) − {J (ξ, a) , J (η, a)} = J ([ξ, η] , Σ (ξ, η)) − {J(ξ) + (ξ, a), J(η) + (η, b)} = J ([ξ, η]) + ([ξ, η] , Σ(ξ, η)) − {J(ξ), J(η)} = Σ(ξ, η) + ([(ξ, a), (η, b)]) = (λ + )([(ξ, a), (η, b)]), (12.3.25) where λ(ξ, a) = a. Thus, the real-valued two-cocycle of the g action is a coboundary and hence J can be adjusted to become infinitesimally equivariant. Thus, J (ξ, a) = J(ξ) − a (12.3.26) is the desired infinitesimally equivariant momentum map of g on P . For example, the action of R2 on itself by translations has the nonequiv- ariant momentum map J(q, p), (ξ, η) = ξp − ηq with group one-cocycle σ(x, y) · (ξ, η) = ξy − ηx; here we think of R2 endowed with the symplectic form dq ∧ dp. The corresponding infinitesimally equivariant momentum map of the central extension is given by (12.3.26), that is, by the expression J (q, p), (ξ, η, a) = ξp − ηq − a. For more examples, see §12.4. Consider the situation for the corresponding action of the central extension G of G on P if G = E, a topological vector space regarded as an abelian Lie group. Then g = E, T ση = ση by linearity of ση , so that Σ(ξ, η) = σ(ξ) · η, with ξ on the right-hand side thought of as an element of the Lie group G. One defines the central extension G of G by the circle group S 1 as the Lie group having an underlying manifold E × S 1 , and whose multiplication is given by (Souriau [1969]) q1 , eiθ1 · q2 , eiθ2 = q1 + q2 , exp i θ1 + θ2 + 1 Σ(q1 , q2 ) 2 , (12.3.27) the identity element equal to (0, 1), and the inverse given by −1 q, eiθ = −q, e−iθ . Then the Lie algebra of G is g = E ⊕R with the bracket given by (12.3.22) and thus the G -action on P given by (q, eiθ ) · z = q · z has an equivariant momentum map J given by (12.3.26). If E = R2 , the group G is the Heisenberg group (see Exercise 9.1-4). ........................... 15 July 1998—18h02 ...........................

424. 410 12. Computation and Properties of Momentum Maps Global Equivariance. Assume J is a Lie algebra homomorphism. Since Γη,g is a Casimir function on P for every g ∈ G and η ∈ g, it follows that Γη |G × S is independent of z ∈ S, where S is a symplectic leaf. Denote this function that depends only on the leaf S by ΓS : G → R. Fixing z ∈ S, η and taking the derivative of the map g → ΓS (g, z) at g = e in the direction η ξ ∈ g, gives −(ad ξ)∗ J(z), η − Tz J · ξP (z), η = 0, (12.3.28) that is, Te ΓS = 0 for all η ∈ g. By Proposition 12.4.1(ii), we have η Γη (gh) = Γη (g) + ΓAdg−1 η (h). (12.3.29) Taking the derivative of (12.3.29) with respect to g in the direction ξ at h = e on the leaf S and using Te ΓS = 0, we get η Tg ΓS (Te Lg (ξ)) = Te ΓS g−1 η (ξ) = 0. η Ad (12.3.30) Thus, Γη is constant on G × S (recall that both G and the symplectic leaves are, by definition, connected). Since Γη (e, z) = 0, it follows that Γη |G × S = 0 for any leaf S and hence Γη = 0 on G × P . But Γη = 0 for every η ∈ g is equivalent to equivariance. Together with Theorem 11.5.1 this proves the following: Theorem 12.3.2. Let the connected Lie group G act canonically on the left on the Poisson manifold P . The action of G is globally Hamiltonian if and only if there is a Lie algebra homomorphism ψ : g → F(P ) such that Xψ(ξ) = ξP for all ξ ∈ g where ξP is the infinitesimal generator of the G-action. If J is the equivariant momentum map of the action, then we can take ψ = J. The converse question of the construction of a group action whose momentum map equals a given set of conserved quantities closed under bracketing is addressed in Fong and Meyer [1975]. See also Vinogradov and Krasilshchick [1975] and Conn [1984], [1985] for the related question of when the germs of Poisson vector fields are Hamiltonian. Exercises 12.3-1. Let G be a Lie group, g its Lie algebra, and g∗ its dual. Let ∧k (g∗ ) be the space of maps α : g∗ × · · · × g∗ (k times ) → R such that α is k-linear and skew-symmetric. Define, for each k ≥ 1, the map d : ∧k (g∗ ) −→ ∧k+1 (g∗ ), ........................... 15 July 1998—18h02 ...........................

425. 12.4 Equivariant Momentum Maps Are Poisson 411 by dα(ξ0 , ξ1 , . . . ξk ) = ˆ ˆ (−1)i+j α([ξi , ξj ], ξ0 , . . . , ξi , . . . ξj , . . . ξk ), 0≤ij≤k ˆ where ξi means that ξi is omitted. (a) Work out dα explicitly, if α ∈ ∧1 (g∗ ) and α ∈ ∧2 (g∗ ). (b) Show that if we identify α ∈ ∧k (g∗ ) with its left invariant extension αL ∈ Ωk (G) given by αL (g)(v1 , . . . vk ) = α(Te Lg−1 v1 , . . . Te Lg−1 vk ), where v1 , . . . , vk ∈ Tg G, then dαL is the left invariant extension of dα, that is, dαL = (dα)L . (c) Conclude that indeed dα ∈ ∧k+1 (g∗ ) if α ∈ ∧k (g∗ ) and that d◦d = 0. (d) Letting Z k (g) = ker d : ∧k (g∗ ) −→ ∧k+1 (g∗ ) be the subspace of k-cocyles and B k ((g) = range d : ∧k−1 (g∗ ) −→ ∧k (g∗ ) be the space of k-coboundaries, show that B k (g) ⊂ Z k (g). The quotient H k (g)/B k (g) is the k-th Lie algebra cohomology group of g with real coefficients. 12.3-2. Compute the group and Lie algebra cocyles for the momentum map of SE(2) on R2 given in Exercise 11.4-3. 12.4 Equivariant Momentum Maps Are Poisson We next show that equivariant momentum maps are Poisson maps. This provides a fundamental method for finding canonical maps between Poisson manifolds. This result is partly contained in Lie’s work [1890], is implicit in Guillemin and Sternberg [1980], and explicit in Holmes and Marsden [1983] and Guillemin and Sternberg [1984]. Theorem 12.4.1 (Canonical Momentum Maps). If J : P → g∗ is an infinitesimally equivariant momentum map for a left Hamiltonian action of g on a Poisson manifold P , then J is a Poisson map: J∗ {F1 , F2 }+ = {J∗ F1 , J∗ F2 } , (12.4.1) ........................... 15 July 1998—18h02 ...........................

426. 412 12. Computation and Properties of Momentum Maps that is, {F1 , F2 }+ ◦ J = {F1 ◦ J, F2 ◦ J} ∗ for all F1 , F2 ∈ F(g ), where { , }+ denotes the “+” Lie–Poisson bracket. Proof. Infinitesimal equivariance means that J([ξ, η]) = {J(ξ), J(η)}. For F1 , F2 ∈ F(g∗ ), let z ∈ P, ξ = δF1 /δµ, and η = δF2 /δµ evaluated at the particular point µ = J(z) ∈ g∗ . Then δF1 δF2 J∗ {F1 , F2 }+ (z) = µ, , δµ δµ = µ, [ξ, η] = J([ξ, η])(z) = {J(ξ), J(η)} (z). But for any z ∈ P and vz ∈ Tz P, d(F1 ◦ J)(z) · vz = dF1 (µ) · Tz J(vz ) δF1 = Tz J(vz ), δµ = dJ(ξ)(z) · vz , that is, (F1 ◦ J)(z) and J(ξ)(z) have equal z-derivatives. Since the Poisson bracket on P depends only on the point values of the first derivatives, we conclude that {F1 ◦ J, F2 ◦ J} (z) = {J(ξ), J(η)} (z). Theorem 12.4.2 (Collective Hamiltonian Theorem). Let J : P → g∗ be a momentum map. Let z ∈ P and µ = J(z) ∈ g∗ . Then for any F ∈ F(g∗ ), + δF XF ◦J (z) = XJ(δF/δµ) (z) = (z). (12.4.2) δµ P Proof. For any H ∈ F(P ), XF ◦J [H](z) = − XH [F ◦ J](z) = −d(F ◦ J)(z) · XH (z) δF = − dF (µ)(Tz J · XH (z)) = − Tz J(XH (z)), δµ δF δF = − dJ (z) · XH (z) = −XH J (z) δµ δµ = XJ(δF/δµ) [H](z). This proves the first equality in (12.4.2) and the second results from the definition of the momentum map. ........................... 15 July 1998—18h02 ...........................

427. 12.4 Equivariant Momentum Maps Are Poisson 413 Functions on P of the form F ◦ J are called collective. Note that if F is the linear function determined by ξ ∈ g, (12.4.2) reduces to XJ(ξ) (z) = ξP (z), the definition of the momentum map. To demonstrate the relation between these results, let us derive Theorem 12.4.1 from Theorem 12.4.2. Let µ = J(z), and F, H ∈ F(g∗ ). Then + δF δH J∗ {F, H}+ (z) = {F, H}+ (J(z)) = J(z), , δµ δµ δF δH δF δH =J , (z) = J ,J (z) δµ δµ δµ δµ (by infinitesimal equivariance) δF δF = XJ(δH/δµ) J (z) = XH◦J J (z) δµ δµ (by the collective Hamiltonian theorem) = −XJ(δF/δµ) [H ◦ J](z) = −XF ◦J [H ◦ J](z) (again by the collective Hamiltonian theorem) = {F ◦ J, H ◦ J} (z). Remarks. 1. Let i : g → F(g∗ ) denote the natural embedding of g in its bidual; that is, i(ξ) · µ = µ, ξ . Since δi(ξ)/δµ = ξ, i is a Lie algebra homomorphism, that is, i([ξ, η]) = {i(ξ), i(η)}+ . (12.4.3) We claim that a canonical left Lie algebra action of g on a Poisson manifold P is Hamiltonian if and only if there is a Poisson algebra homomorphism χ : F(g∗ ) → F(P ) such that X(χ◦i)(ξ) = ξP for all ξ ∈ g. Indeed, if + the action is Hamiltonian, let χ = J∗ (pull back on functions) and the assertion follows from the definition of momentum maps. The converse relies on the following fact. Let M, N be finite dimensional manifolds and χ : F(N ) → F(M ) be a ring homomorphism. Then there exists a unique smooth map ϕ : M → N such that χ = ϕ∗ . (A similar statement holds for infinite-dimensional manifolds in the presence of some additional technical conditions. See Abraham, Marsden, and Ratiu [1988], Supplement 4.2C.) Therefore, if a ring and Lie algebra homomorphism F(g∗ ) → F(P ) is given, + there is a unique map J : P → g∗ such that χ = J∗ . But for ξ, µ ∈ g∗ we have [(χ ◦ i)(ξ)](z) = J∗ (i(ξ))(z) = i(ξ)(J(z)) = J(z), ξ = J(ξ)(z), (12.4.4) ........................... 15 July 1998—18h02 ...........................

428. 414 12. Computation and Properties of Momentum Maps that is, χ ◦ i = J which is a Lie algebra homomorphism because χ is, by hypothesis. Since XJ(ξ) = ξP again by hypothesis, it follows that J is an infinitesimally equivariant momentum map. 2. Here we have worked with left actions. If in all statements one changes left by right actions and “+” by “−” in the Lie–Poisson structures on g∗ , the resulting statements are true. Examples (a) Phase Space Rotations. Let (P, Ω) be a linear symplectic space and let G be a subgroup of the linear symplectic group acting on P by matrix multiplication. The infinitesimal generator of ξ ∈ g at z ∈ P is ξP (z) = ξz, (12.4.5) where ξz is matrix multiplication. This vector field is Hamiltonian with Hamiltonian Ω(ξz, z)/2 by Proposition 2.7.1. Thus, a momentum map is J(z), ξ = 1 Ω(ξz, z). 2 (12.4.6) For S ∈ G, the adjoint action is AdS ξ = SξS −1 , (12.4.7) and hence 1 J(Sz), SξS −1 = Ω(SξS −1 Sz, Sz) 2 1 1 = Ω(Sξz, Sz) = Ω(ξz, z), (12.4.8) 2 2 so J is equivariant. Infinitesimal equivariance is a reformulation of (2.7.10). Notice that this momentum map is not of the cotangent lift type. (b) Phase Space Translations. Let (P, Ω) be a linear symplectic space and let G be a subgroup of the translation group of P , with g identified with a linear subspace of P . Clearly ξP (z) = ξ in this case. The vector field is Hamiltonian with Hamiltonian given by the linear function J(ξ)(z) = Ω(ξ, z), (12.4.9) as is easily checked. This is therefore a momentum map for the action. This momentum map is not equivariant, however. The action of R2 on R2 by translation is a specific example; see the end of Remark 3 of §12.3. ........................... 15 July 1998—18h02 ...........................

429. 12.4 Equivariant Momentum Maps Are Poisson 415 (c) Lifted Actions and Magnetic Terms. Another way nonequivariance of momentum maps comes up is with lifted cotangent actions, but with symplectic forms which are the canonical ones modified by the addition of a magnetic term. For example, endow P = T ∗ R2 with the symplectic form ΩB = dq 1 ∧ dp1 + dq 2 ∧ dp2 + B dq 1 ∧ dq 2 where B is a function of q 1 and q 2 . Consider the action of R2 on R2 by translations and lift this to an action of R2 on P . Note that this action preserves ΩB if and only if B is constant, which will be assumed from now on. By (12.4.9) the momentum map is J(q, p), ξ = p · ξ + B(ξ 1 q 2 − ξ 2 q 1 ). (12.4.10) This momentum map is not equivariant; in fact, since R2 is abelian, its Lie algebra two-cocycle is given by Σ(ξ, η) = −{J(ξ), J(η)} = −2B(ξ 1 η 2 − ξ 2 η 1 ). Let us assume from now on that B is nonzero. Viewed in different coordinates, the form ΩB can be made canonical and the action by R2 is still translation by a canonical transformations. To do this, one switches to guiding center coordinates (R, P) defined by P = p and R = (q 1 − p2 /B, q 2 + p1 /B). The physical interpretation of these coordinates is the following: P is the momentum of the particle, while R is the center of the nearly circular orbit pursued by the particle with coordinates (q, p) when the magnetic field is strong (Littlejohn [1983, 1984]). In these coordinates, ΩB takes the form 1 ΩB = BdR1 ∧ dR2 − dP1 ∧ dP2 B and the R2 -action on T ∗ R2 becomes translation in the R-variable. The momentum map (12.4.10) becomes J(R, P), ξ = B(ξ 1 R2 − ξ 2 R1 ) (12.4.11) which is again a special case of (12.2.5). The cohomology class [Σ] = 0, as the following argument shows. If Σ was exact, there would exist a linear functional λ : R2 → R such that Σ(ξ, η) = λ([ξ, η]) = 0 for all ξ, η; this is clearly false. Thus, J cannot be adjusted to obtain an equivariant momentum map. Following Remark 5 of §12.3, the nonequivariance of the momentum map can be removed by passing to a central extension of R2 . Namely, let G = R2 × S 1 with multiplication given by b −a2 b1 )) 1 2 (a, eiθ )(b, eiϕ ) = a + b, ei(θ+ϕ+B(a (12.4.12) ........................... 15 July 1998—18h02 ...........................

430. 416 12. Computation and Properties of Momentum Maps and letting G act on T ∗ R2 as before by (a, eiθ ) · (q, p) = (q + a, p). Then the momentum map J : T ∗ R2 → g ∗ = R3 is given by J(q, p), (ξ, a) = p · ξ + B(ξ 1 q 2 − ξ 2 q 1 ) − a. (12.4.13) (d) Clairaut’s Theorem. Let M be a surface of revolution in R3 obtained by revolving a graph r = f (z) about the z-axis, where f is a smooth positive function. Pull back the usual metric of R3 to M and note that it is invariant under rotations about the z-axis. Consider the geodesic flow on M . The momentum map associated with the S 1 symmetry is J : T M → R given by J(q, v), ξ = (q, v), ξM (q) , as usual. Here, ξM is the vector field on R3 associated with a rotation with angular velocity ξ about the z-axis, so ξM (q) = ξk × q . Thus, J(q, v), ξ = ξr v cos θ, where r is the distance to the z-axis and θ is the angle between v and the horizontal plane. Thus, as v is conserved, by conservation of energy, r cos θ is conserved along any geodesic on a surface of revolution, a statement known as Clairaut’s Theorem. (e) Mass of a nonrelativistic free quantum particle. Here we show by means of an example, the relation between (genuine) projective unitary representations and non equivariance of the momentum map for the action on the projective space. This complements the discussion in Example (m) of §12.2 where we have shown that for unitary representations the momentum map is equivariant. Let G be the Galilean group introduced in Example (c) following Proposi- tion 9.3.10, that is the subgroup of GL(5, R) consisting of matrices   R v a g=0 1 τ 0 0 1 where R ∈ SO(3), v, a ∈ R3 , and τ ∈ R. Let H = L2 (R3 ; C) be the Hilbert space of square (Lebesgue) integrable complex functions on R3 . Fix a real number m = 0; for each g = {R, v, a, τ } ∈ G, define the following unitary operator in H: τ (Um (g)f )(p) = exp i |p|2 + (p + mv) · a f (R−1 (p + mv)). 2m (12.4.14) ........................... 15 July 1998—18h02 ...........................

431. 12.4 Equivariant Momentum Maps Are Poisson 417 We can check by direct computation that: Um (g1 )Um (g2 ) = exp(−imσ(g1 , g2 ))Um (g1 g2 ), (12.4.15) where (with gj = {Rj , vj , aj , τj }) σ(g1 , g2 ) = 1 |v1 |2 τ2 + (R1 v2 ) · (v1 τ2 + a1 ). 2 (12.4.16) Note that σ(e, g) = σ(g, e) = 0, σ(g, g −1 ) = σ(g −1 , g), and Um (g −1 ) = exp(−imσ(g, g −1 ))Um (g)−1 . From (12.4.15), we see that the map g → Um (g) is not a group homomorphism, because of an overall factor in S 1 . Clearly, eiφ → eiφ f is a unitary operator on H = L2 (R3 ; C) and is a normal subgroup of U(H). Define the projective unitary group of H by U(PH) = U(H)/S 1 . Then (12.4.15) induces a group homomorphism g ∈ G → [Um (g)] ∈ U(PH), that is, we have a projective unitary representation of the Galilean group on H = L2 (R3 ; C). It is easy to see that this action of the Galilean group G on PH is symplectic (use the formula in Proposition (5.3.1)). Prop ref. needs Next, we compute the infinitesimal generators of this action. Note that to be checked for any smooth f ∈ H = L2 (R3 ; C), the map g → Um (g)f is also smooth, so D := C ∞ (R3 ; C) is invariant under the group action. Thus, it makes sense to define for any f ∈ D, (a(ξ))f = Te (Um (·)f ) · ξ, (12.4.17) where e is the identity matrix in G and ξ ∈ g is arbitrary. This formula shows that a(ξ) is linear in ξ, thereby defining a linear operator a : D = C ∞ (R3 ; C) → H = L2 (R3 ; C). Because Um (g) is unitary and Um (e) = identity operator on H, it follows that a(ξ) is formally skew-adjoint on D for any ξ ∈ g. Explicitly, if   ˆ ω u α ξ=0 0 θ 0 0 0 (see Example (c) following Proposition 9.3.10), we get θ ∂f (a(ω)f )(p) = i |p|2 + p · α f (p) + (mu − ω × p) · (12.4.18) 2m ∂p or, expressed as a collection of four operators corresponding to ω, u, α, and θ, ∂f ∂f (a(ω)f )(p) = −ω · p × , (a(u)f )(p) = mu · , ∂p ∂p |p|2 (a(α)f )(p) = i(α · p)f (p), (a(θ)f )(p) = iθ f (p). 2m ........................... 15 July 1998—18h02 ...........................

432. 418 12. Computation and Properties of Momentum Maps From these formulas we see that a(ξ)f is well defined for f ∈ D and that D is invariant under all a(ξ) for ξ ∈ D. Thus, a(ξ) is uniquely determined as an unbounded skew adjoint operator on H. Stone’s Theorem (see MTA) guarantees that [exp ta(ξ)]f = Um (exp tξ)f (12.4.19) is C ∞ in t with derivative at t = 0 equal to a(ξ)f , for all f ∈ D. Clearly, the obvious formulas (taking equivalence classes) defines a(ξ) on PD and hence conditions (i), (ii), and (iii) of Example (f) in §9.3 hold; therefore PD is an essential G-smooth part of PH. The momentum map of the projective unitary representation of the Galilean group G on PH can thus be defined on PD. By Example (g) of §11.5, this momentum map is induced from that now it is (g) of the G-action on H and has thus the expression but I will insert there(?) i f, a(ξ)f other things, J(ξ)([f ]) = − (12.4.20) 2 f 2 so it needs to be checked for f = 0. later In spite of the fact that (12.4.20) and (11.4.24) look practically the same, the corresponding momentum maps have different properties because the infinitesimal generators a(ξ) behave differently from A(ξ): in (11.4.24), A(ξ) is uniquely determined by ξ, but here a(ξ) is given by the projective representation only up to a linear functional on g. More crucial, the equivariance relation (12.2.20), which holds for the unitary representation, fails for projective representation. Indeed, let us show that a(Adg ξ) = Um (g)a(ξ)Um (g)−1 + 2iΓξ (g −1 )1H , (12.4.21) where 1H is the identity operator on H and Γξ (g −1 ) ∈ R is a number Wendy, funny explicitly computed below. To show this, note that from (12.4.19) and scriptscript fonts elimi- (12.4.15) we get nated eta(Adg ξ) = Um (exp t Adg ξ) = Um (g(exp tξ)g −1 ) = Um (g)Um (exp tξ)Um (g)−1 exp(imγ(g, tξ)), (12.4.22) where γ(g, tξ) = σ(g, (exp tξ)g −1 ) + σ(exp tξ, g −1 ) − σ(g, g −1 ). (12.4.23) Note that γ(g, 0) = 0. Taking the derivative of (12.4.22) with respect to t at t = 0 and using Stone’s theorem, we get (12.4.21) with m d Γξ (g −1 ) = γ(g, tξ) . (12.4.24) 2 dt t=0 ........................... 15 July 1998—18h02 ...........................

433. 12.4 Equivariant Momentum Maps Are Poisson 419 Using the notations in §9.3, (12.4.16), and (12.4.23), we have for ξ = {ω, u, α, θ} and g = {R, v, a, τ }: m 1 Γξ (g −1 ) = − |v|2 θ + (Rω) · (a × v) + a · Ru − v · Rα . (12.4.25) 2 2 which implies, using   R−1 −R−1 v R−1 (τ v − a) g −1 = 0 1 −τ  0 0 1 that m 1 Γξ (g) = − |v|2 θ + ω · (a × v) + (τ v − a) · u + v · α . (12.4.26) 2 2 The g∗ -valued group one-cocyle deﬁned by the momentum map (12.4.20) is thus given by J(ξ)(g · [f ]) − J(Adg−1 ξ)([f ]) = Γξ (g), in agreement with the notation of Proposition 12.3.1. The real-valued Lie algebra two-cocycle is thus given by (see 12.3.11) d Σ(ξ, η) = Te Γη (ξ) = Γη (c(t)) dt t=0 m = (u · α − u · α), (12.4.27) 2 where ξ = {ω, u, α, θ}, η = {ω , u , α , θ }, and c(t) = {etω , tu, tα, tθ}. ˆ This cocycle on the Lie algebra is nontrivial, that is, its cohomology class is non zero (see Exercise 12.4-6). Therefore, the mass of the particle measures the obstruction to equivariance for the momentum map (or for the projective representation to be a unitary representation) in H 2 (g, R). Exercises 12.4-1. Verify directly that angular momentum is a Poisson map. 12.4-2. What does the collective Hamiltonian theorem state for angular momentum? Is the result obvious? 12.4-3. If z(t) is an integral curve of XF ◦J , show that µ(t) = J(z(t)) satisﬁes µ = ad∗ ˙ δF/δµ µ. 12.4-4. Consider an ellipsoid of revolution in R3 and a geodesic starting at the “equator” making an angle of α with the equator. Use Clairaut’s theorem to derive a bound on how high the geodesic climbs up the ellipse. ........................... 15 July 1998—18h02 ...........................

434. 420 12. Computation and Properties of Momentum Maps 12.4-5. Consider the action of SE(2) on R2 as described in Exercise 11.4- 3. Since this action was not defined as a lift, Theorem 12.1.4 is not applicable. In fact, in Exercise 11.6-2 it was shown that this momentum map is not equivariant. Compute the group and Lie algebra cocycles defined by this momentum map. Find the Lie algebra central extension making the momentum map equivariant. 12.4-6. Using Exercise 12.4-1, show that for the Galilean algebra, any 2-coboundary has the form: λ(ξ, ξ ) = x · (ω × ω ) + y · (ω × u − ω × u) + z · (ω × α − ω × α + uθ − u θ), for some x, y, z ∈ R3 , where ξ = {ω, u, α, θ} and ξ = {ω , u , α , θ }. Conclude that the cocyle Σ in Example (e) (see 12.4.27) is not a coboundary. (It can be proven that H 2 (g, R) ∼ R, that is, it is 1-dimensional, but = this requires more algebraic work (Gullemin and Sternberg [1977,1984])). 12.4-7. Deduce the formula for the momentum map in Exercise 11.5-4 from (12.4.6) given in Example (a). Is this the right x-ref? 12.5 Poisson Automorphisms Here are some miscellaneous facts about Poisson automorphisms, symplectic leaves, and momentum maps. For a Poisson manifold P , define the following Lie subalgebras of X(P ): • Infinitesimal Poisson Automorphisms. Let P(P ) be the set of X ∈ X(P ) such that: X[{F1 , F2 }] = {X[F1 ], F2 } + {F1 , X[F2 ]}. • Infinitesimal Poisson Automorphisms Preserving Leaves. Let PL(P ) be the set of X ∈ P(P ) such that X(z) ∈ Tz S, where S is the symplectic leaf containing z ∈ P. • Locally Hamiltonian Vector Fields Let LH(P ) be the set of X ∈ X(P ) such that for each z ∈ P, there is an open neighborhood U of z and an F ∈ F(U ) such that X|U = XF |U. • Hamiltonian Vector Fields. Let H(P ) be the set of Hamiltonian vector fields XF for F ∈ F(P ). Then one has the following facts (references are given if the verification is not straightforward): ........................... 15 July 1998—18h02 ...........................

435. 12.6 Momentum Maps and Casimir Functions 421 1. H(P ) ⊂ LH(P ) ⊂ PL(P ) ⊂ P(P ). 2. If P is symplectic, then LH(P ) = PL(P ) = P(P ) and if H 1 (P ) = 0, then LH(P ) = H(P ). 3. Let P be the trivial Poisson manifold, that is, {F, G} = 0 for all F, G ∈ F(P ). Then P(P ) = PL(P ). 4. Let P = R2 with the bracket ∂F ∂G ∂F ∂F {F, G}(x, y) = x − . ∂x ∂y ∂x ∂y This is, in fact, a Lie–Poisson bracket. The vector ﬁeld ∂ X(x, y) = xy ∂y is an example of an element of PL(P ) which is not in LH(P ). 5. H(P ) is an ideal in any of the three Lie algebras including it. Indeed, if Y ∈ P(P ) and H ∈ F(R), then [Y, XH ] = XY [H] . 6. If P is symplectic, then [LH(P ), LH(P )] ⊂ H(P ). (The Hamiltonian for [X, Y ] is −Ω(X, Y ).) This is false for Poisson manifolds in general. If P is symplectic, Calabi [1970] and Lichnerowicz [1973] showed that [LH(P ), LH(P )] = H(P ). 7. If the Lie algebra g admits a momentum map on P , then gP ⊂ H(P ). 8. Let G be a connected Lie group. If the action admits a momentum map, it preserves the leaves of P . The proof was given in §12.4. 12.6 Momentum Maps and Casimir Functions In this section we return to Casimir functions studied in Chapter 10 and link them with momentum maps. We will do this in the context of the Poisson manifolds P/G studied in §10.7. We start with a Poisson manifold P and a free and proper Poisson action of a Lie group G on P admitting an equivariant momentum mapping J : P → g∗ . We want to link J with a Casimir function C : P/G → R. Proposition 12.6.1. Let Φ : g∗ → R be a function that is invariant under the coadjoint action. Then: (i) Φ is a Casimir function for the Lie–Poisson bracket; ........................... 15 July 1998—18h02 ...........................

436. 422 12. Computation and Properties of Momentum Maps (ii) Φ ◦ J is G-invariant on P and so defines a function C : P/G → R such that Φ ◦ J = C ◦ π, as in Figure 12.8.1; and (iii) the function C is a Casimir function on P/G. P d π d J d © ‚ d P/G g∗ d d C d Φ ‚ © d R Figure 12.6.1. Casimir functions and momentum maps. Proof. To prove the first part, we write down the condition of Ad∗ - invariance as Φ(Ad∗−1 µ) = Φ(µ). g (12.6.1) Differentiate this relation with respect to g at g = e in the direction η to get (see equation (9.3.1)), d 0= Φ Ad∗ ∗ exp(−tη) µ = −DΦ(µ) · adη µ, (12.6.2) dt t=0 for all η ∈ g. Thus, by definition of δΦ/δµ, δΦ δΦ 0= ad∗ µ, η = µ, adη = − ad∗ δΦ/δµ µ, η δµ δµ for all η ∈ g. In other words, ad∗ δΦ/δµ µ = 0 so by Proposition 10.7.1, XΦ = 0 and thus Φ is a Casimir function. To prove the second part, note that, by equivariance of J and invariance of Φ, Φ(J(g · z)) = Φ(Ad∗−1 J(z)) = Φ(J(z)), g so Φ ◦ J is G-invariant. Finally, for the third part, we use the collective Hamiltonian Theorem 12.4.2 to get for µ = J(z), δΦ XΦ◦J (z) = (z) δµ P ........................... 15 July 1998—18h02 ...........................

437. 12.6 Momentum Maps and Casimir Functions 423 and so Tz π·XΦ◦J (z) = 0 since infinitesimal generators are tangent to orbits, so project to zero under π. But π is Poisson, so 0 = Tz π · XΦ◦J (z) = Tz π · XC◦π (z) = XC (π(z)). Thus, C is a Casimir function on P/G. Corollary 12.6.2. If G is Abelian and Φ : g∗ → R is any smooth function, then Φ ◦ J = C ◦ π defines a Casimir function C on P/G. This follows because for Abelian groups, the Ad∗ -action is trivial, so any function on g∗ is Ad∗ -invariant. Exercises 12.6-1. Verify that Φ(Π) = Π 2 is an invariant function on so(3)∗ . 12.6-2. Use Corollary 12.6.2 to find the Casimir functions for the bracket (10.5.6). 12.6-3. Show that a left invariant Hamiltonian H : T ∗ G → R collec- tivizes relative to the momentum map for the right action, but need not collectivize for the momentum map of the left action. ........................... 15 July 1998—18h02 ...........................

438. 424 12. Computation and Properties of Momentum Maps ........................... 15 July 1998—18h02 ...........................

439. This is page 425 Printer: Opaque this 13 Lie–Poisson and Euler–Poincar´ e Reduction Besides the Poisson structure on a symplectic manifold, the Lie–Poisson bracket on g∗ , the dual of a Lie algebra, is perhaps the most fundamental example of a Poisson structure. We shall obtain it in the following manner. Given two smooth functions F, H ∈ F(g∗ ), we extend them to functions, FL , HL (respectively, FR , HR ) on all T ∗ G by left (respectively, right) translations. The bracket {FL , HL } (respectively, {FR , HR }) is taken in the canonical symplectic structure Ω on T ∗ G. The result is then restricted to g∗ regarded as the cotangent space at the identity; this deﬁnes {F, H}. We shall prove that one gets the Lie–Poisson bracket this way. This process is called Lie–Poisson reduction. In §14.6 we show that the symplectic leaves of this bracket are the coadjoint orbits in g∗ . There is another side to the story, where the basic objects that are reduced are not Poisson brackets, but rather are variational principles. This aspect, which takes place on g rather than on g∗ , will be told as well. The passage of a variational principle from T G to g is called Euler–Poincar´ e reduction. 13.1 The Lie–Poisson Reduction Theorem We begin by studying the way the canonical Poisson bracket on T ∗ G is related to the Lie–Poisson bracket on g∗ . Theorem 13.1.1 (The Lie–Poisson Reduction Theorem). Identify- ing the set of functions on g∗ with the set of left (respectively, right) in-

440. 426 13. Lie–Poisson and Euler–Poincar´ Reduction e variant functions on T ∗ G endows g∗ with Poisson structures given by δF δH {F, H}± (µ) = ± µ, , . (13.1.1) δµ δµ The space g∗ with this Poisson structure is denoted g∗ (respectively, g∗ ). − + In contexts where the choice of left or right is clear, we shall drop the “ −” or “+” from {F, H}− and {F, H}+ . Following Marsden and Weinstein [1983], this bracket on g∗ is called the Lie–Poisson bracket after Lie [1890], p. 204, where the bracket is given explicitly. See Weinstein [1983a] and §13.7 below for more historical information. In fact, there are already some hints of this structure in Jacobi [1866], p. 7. It was rediscovered several times since Lie’s work. For example, it appears explicitly in Berezin [1967]. It is closely related to results of Arnold, Kirillov, Kostant, and Souriau in the 1960s. Some Terminology. Before proving the theorem, we explain the terminology used in its statement. First, recall from Chapter 9 how the Lie algebra of a Lie group G is constructed. We define g = Te G, the tangent space at the identity. For ξ ∈ g, we define a left invariant vector field ξL = Xξ on G by setting ξL (g) = Te Lg · ξ (13.1.2) where Lg : G → G denotes left translation by g ∈ G and is defined by Lg h = gh. Given ξ, η ∈ g, define [ξ, η] = [ξL , ηL ](e), (13.1.3) where the bracket on the right-hand side is the Jacobi–Lie bracket on vector fields. The bracket (13.1.3) makes g into a Lie algebra, that is, [ , ] is bilinear, antisymmetric, and satisfies Jacobi’s identity. For example, if G is a subgroup of GL(n), the group of invertible n × n matrices, we identify g = Te G with a vector space of matrices and then, as we calculated in Chapter 9, [ξ, η] = ξη − ηξ (13.1.4) is the usual commutator of matrices. A function FL : T ∗ G → R is called left invariant if, for all g ∈ G, FL ◦ T ∗ Lg = FL , (13.1.5) where T ∗ Lg denotes the cotangent lift of Lg , so T ∗ Lg is the pointwise adjoint of T Lg . Let FL (T ∗ G) denote the space of all smooth left invariant functions on T ∗ G. One similarly defines right invariant functions on T ∗ G and the space FR (T ∗ G). Given F : g∗ → R and αg ∈ T ∗ G, set ∗ FL (αg ) = F (Te Lg · αg ) = (F ◦ JR )(αg ), (13.1.6) ........................... 15 July 1998—18h02 ...........................

441. 13.1 The Lie–Poisson Reduction Theorem 427 where JR : T ∗ G → g∗ , JR (αg ) = Te Lg · αg is the momentum map of the ∗ lift of right translation on G (see (12.2.8). The function FL = F ◦ JR is called the left invariant extension of F from g∗ to T ∗ G. One similarly defines the right invariant extension by ∗ FR (αg ) = F (Te Rg · αg ) = (F ◦ JL )(αg ), (13.1.7) where JL : T ∗ G → g∗ , JL (αg ) = Te Rg · αg is the momentum map of the ∗ lift of left translation on G (see (12.2.7)). Right composition with JR (respectively, JL ) thus defines an isomorphism F(g∗ ) → FL (T ∗ G) ( respectively, F(g∗ ) → FR (T ∗ G)) whose inverse is restriction to the fiber Te G = g∗ . ∗ Since T Lg and T Rg are symplectic maps on T ∗ G, it follows that ∗ ∗ FL (T ∗ G) and FR (T ∗ G) are closed under the canonical Poisson bracket on T ∗ G. Thus, one way of rephrasing the Lie–Poisson reduction theorem (we will see another way, using quotients, in §13.4) is to say that the above isomorphisms of F(g∗ ) with FL (T ∗ G) and FR (T ∗ G) respectively, are also isomorphisms of Lie algebras, that is, the following formulas are valid. {F, H}− = {FL , HL }|g∗ (13.1.8) and {F, H}+ = {FR , HR }|g∗ , (13.1.9) where { , }± is the Lie–Poisson bracket on g∗ and { , } is the canonical bracket on T ∗ G. Proof of the Lie–Poisson Reduction Theorem. The map JR : T ∗ G → g∗ − is a Poisson map by Theorem 12.4.1. Therefore, {F, H}− ◦ JR = {F ◦ JR , H ◦ JR } = {FL , HL }. Restriction of this relation to g∗ gives (13.1.8) One similarly proves (13.1.9) using the Poisson property of the map JL : T ∗ G → g∗ . + The proof above was a posteriori , that is, one had to “already know” the formula for the Lie–Poisson bracket. In §13.4 we will prove this theorem again using momentum functions and quotienting by G (see §10.7). This will represent an a priori proof, in the sense that the formula for the Lie– Poisson bracket will be deduced as part of the proof. To gain further insight into this, the next two sections will give constructive proofs of this theorem, in special cases. ........................... 15 July 1998—18h02 ...........................

442. 428 13. Lie–Poisson and Euler–Poincar´ Reduction e Exercises 13.1-1. Let u, v ∈ R3 and deﬁne F u : so(3)∗ R3 → R by Fu (x) = ∗ x, u and similarly for F . Let FL : T SO(3) → R be the left invari- v u ant extension of F u and similarly for FL . Compute the Poisson bracket v {FL , FL }. u v 13.2 Proof of the Lie–Poisson Reduction Theorem for GL(n) We now prove the Lie–Poisson reduction theorem for the special case of the Lie group G = GL(n) of real invertible n × n matrices. Left translation by U ∈ G is given by matrix multiplication: LU A = UA. Identify the tangent space to G at A with the vector space of all n × n matrices, so for B ∈ TA G, TA LU · B = UB as well, since LU A is linear in A. The cotangent space is identiﬁed with the tangent space via the pairing π, B = trace(π T B), (13.2.1) where π T is the transpose of π. The cotangent lift of LU is thus given by T ∗ LU π, B = π, T LU · B = trace(π T UB); that is, T ∗ LU π = UT π. (13.2.2) Given functions F, G : g∗ → R, let FL (A, π) = F (AT π) and GL (A, π) = G(AT π) (13.2.3) be their left invariant extensions. By the chain rule, letting µ = AT π, we get DA FL (A, π) · δA = DF (AT π) · (δA)T π δF = , (δA)T π δµ δF = trace π T δA . (13.2.4) δµ The canonical bracket is therefore δFL δGL δGL δFL {FL , GL } = , − , δA δπ δA δπ δGL δFL = DA FL (A, π) · − DA GL (A, π) · . (13.2.5) δπ δπ ........................... 15 July 1998—18h02 ...........................

443. 13.3 Proof of the Lie–Poisson Reduction Theorem for Diff vol (M ) 429 Since δFL /δπ = δF/δµ at the identity A = Id, where π = µ, using (13.2.4), the Poisson bracket (13.2.5) becomes δG δF δF δG {FL , GL } (µ) = trace µT − µT δµ δµ δµ δµ δF δG δG δF = − µ, − δµ δµ δµ δµ δF δG = − µ, , , (13.2.6) δµ δµ which is the (−)Lie–Poisson bracket. This derivation can be adapted for other matrix groups, including the rotation group SO(3) as special cases. However, in the latter case, one has to be extremely careful to treat the orthogonality constraint properly. Exercises 13.2-1. Let FL and GL have the form (13.2.3) so that it makes sense to restrict FL and GL to T ∗ SO(3). Is the bracket of their restrictions given by the restriction of (13.2.5)? 13.3 Proof of the Lie–Poisson Reduction Theorem for Diff vol (M ) Another special case is G = Diff vol (Ω), the subgroup of the group of diffeomorphisms Diff(Ω) of a region Ω ⊂ R3 , consisting of the volume-preserving diffeomorphisms. We shall treat Diff(Ω) and Diff vol (Ω) formally, although it is known how to handle the functional analysis issues involved (see Ebin and Marsden [1970] and Adams, Ratiu, and Schmid [1986a,b] and references therein). We shall prove (13.1.9) for this case. See the internet supplements of the proof for Diff can (P ). The Lie Algebra of Diff. For η ∈ Diff(Ω), the tangent space at η is given by the set of maps V : Ω → T Ω satisfying V (X) ∈ Tη(X) Ω, that is, vector fields over η. We think of V as a material velocity field. Thus, the tangent space at the identity is the space of vector fields on Ω (tangent to ∂Ω). Given two such vector fields, their left Lie algebra bracket is related to the Jacobi–Lie bracket by (see Chapter 9): [V, W ]LA = − [V, W ]JL , that is, [V, W ]LA = (W · )V − (V · )W, (13.3.1) ........................... 15 July 1998—18h02 ...........................

444. 430 13. Lie–Poisson and Euler–Poincar´ Reduction e as one finds using the definitions. Right Translation. We will be computing the right Lie–Poisson bracket on g∗ . Right translation by ϕ on G is given by Rϕ η = η ◦ ϕ. (13.3.2) Differentiating (13.3.2) with respect to η gives T Rϕ · V = V ◦ ϕ. (13.3.3) Identify Tη G with those V ’s such that the vector field on R3 given by v = V ◦ η −1 , is divergence-free and identify Tη G with Tη G via the pairing ∗ π, V = π · V dx dy dz, (13.3.4) Ω where π · V is the dot product on R3 . By the change of variables formula, and the fact that ϕ ∈ G has unit Jacobian, T ∗ Rϕ · π, V = π, T Rϕ · V = π · (V ◦ ϕ) dx dy dz = (π ◦ ϕ−1 ) · V dx dy dz, Ω Ω so T ∗ Rϕ · π = π ◦ ϕ−1 . (13.3.5) Derivatives of Right Invariant Extensions. If F : g∗ → R is given, its right invariant extension is FR (η, π) = F (π ◦ η −1 ). (13.3.6) Let us denote elements of g∗ by M, so we are investigating the relation between the canonical bracket of FR and HR and the Lie–Poisson bracket of F and H via the relation M ◦ η = π. From (13.3.6) and the chain rule, we get Dη FR (Id, π) · v = −DM F (M) · Dη π(Id) · v δF = − ((v · )M) · dx dy dz, (13.3.7) Ω δM where δF/δM is a divergence-free vector field parallel to the boundary. Since T ∗ G is not given as a product space, one has to worry about what it means to hold π constant in (13.3.7). We leave it to the ambitious reader to justify this formal calculation. ........................... 15 July 1998—18h02 ...........................

445. 13.4 Lie–Poisson Reduction using Momentum Functions 431 Computation of Brackets. Thus, the canonical bracket at the identity becomes δFR δHR δHR δFR {FR , HR } (Id, π) = − dx dy dz Ω δη δπ δη δπ δHR δFR = Dη FR (Id, π) · − Dη HR (Id, π) · . (13.3.8) δπ δπ At the identity, π = M and δFR /δπ = δF/δM, so substituting this and (13.3.7) into (13.3.8), we get {FR , HR }(Id, M) δH δF δF δH =− · M· − · M· dx dy dz. (13.3.9) Ω δM δM δM δM Equation (13.3.9) may be integrated by parts to give {FR , HR } (Id, M) δH δF δF δH = M· · − · dx dy dz δM δM δM δM δF δH = M· , dx dy dz, (13.3.10) δM δM LA which is the “+” Lie–Poisson bracket. In doing this step note div(δH/δM) = 0 and since δH/δM and δF/δM are parallel to the boundary, no boundary term appears. When doing free boundary problems, these boundary terms are essential to retain (see Lewis, Marsden, Montgomery, and Ratiu [1986]). For other diffeormorphism groups, it may be convenient to treat M as a one-form density rather than a vector field. 13.4 Lie–Poisson Reduction using Momentum Functions Identifiying the Quotient as g∗ . Now we turn to a constructive proof of the Lie–Poisson reduction theorem using momentum functions. We begin by observing that T ∗ G/G is diffeomorphic to g∗ . To see this, note that the trivialization of T ∗ G by left translations given by λ : αg ∈ Tg G → (g, Te Lg (αg )) = (g, JR (αg )) ∈ G × g∗ ∗ ∗ transforms the usual cotangent lift of left translation on G into the G-action on G × g∗ given by g · (h, µ) = (gh, µ), (13.4.1) ........................... 15 July 1998—18h02 ...........................

446. 432 13. Lie–Poisson and Euler–Poincar´ Reduction e for g, h ∈ G and µ ∈ g∗ . Therefore, T ∗ G/G is diffeomorphic to (G × g∗ )/G which in turn equals g∗ , since G does not act on g∗ (see (13.4.1)). Thus, we can regard JR : T ∗ G → g∗ as the canonical projection T ∗ G → T ∗ G/G and, as a consequence of the Poisson Reduction Theorem (Chapter 10), g∗ inherits a Poisson bracket, which we will call { , }− for the time being, uniquely characterized by the relation: {F, H}− ◦ JR = {F ◦ JR , H ◦ JR } (13.4.2) for any functions F, H ∈ F(g∗ ). The goal of this section is to explicitly compute this bracket { , }− and to discover at the end that it equals the (−) Lie–Poisson bracket. Before beginning the proof, it is useful to recall that the Poisson bracket {F, H}− , for F, H ∈ F(g∗ ) depends only on the differentials of F and H at each point. Thus, in determining the bracket { , }− on g∗ , it is enough to assume that F and H are linear functions on g∗ . Proof of the Lie–Poisson Reduction Theorem. The space FL (T ∗ G) of left invariant functions on T ∗ G is isomorphic (as a vector space) to F(g∗ ), the space of all functions on the dual g∗ of the Lie algebra g of G. This isomorphism is given by F ∈ F(g∗ ) ↔ FL ∈ FL (T ∗ G), where ∗ FL (αg ) = F (Te Lg · αg ). (13.4.3) Since FL (T ∗ G) is closed under bracketing (which follows because T ∗ Lg is a symplectic map), F(g∗ ) gets endowed with a unique Poisson structure. As we remarked just above, it is enough to consider the case in which F is replaced by its linearization at a particular point. This means, it is enough to prove the Lie–Poisson reduction theorem for linear functions on g∗ . If F is linear, we can write F (µ) = µ, δF/δµ , where δF/δµ is a constant in g, ∗ so that letting µ = Te Lg · αg , we get ∗ ∗ δF FL (αg ) = F (Te Lg · αg ) = Te Lg · αg , δµ δF δF = αg , Te Lg · =P (αg ), (13.4.4) δµ δµ L where ξL (g) = Te Lg (ξ) is the left invariant vector field on G whose value at e is ξ ∈ g. Thus, by (12.1.2), (13.4.4), and the definition of the Lie algebra ........................... 15 July 1998—18h02 ...........................

447. 13.5 Reduction and Reconstruction of Dynamics 433 bracket, we have δF δH {FL , HL } (µ) = P ,P (µ) δµ L δµ L δF δH = −P , (µ) δµ L δµ L δF δH = −P , (µ) δµ δµ L δF δH = − µ, , , (13.4.5) δµ δµ as required. Since F ◦ JR = FL and H ◦ JR = HL , formulas (13.4.2) and (13.4.6) give ∂F ∂H {F, H}− (µ) = {FL , HL }(µ) = − µ, , , ∂µ ∂µ that is, the bracket { , }− introduced via identifying T ∗ G/G with g∗ equals the (−) Lie–Poisson bracket. The formula with “+” follows in a similar way by making use of right invariant extensions of linear functions since the Lie bracket of two right invariant vector ﬁelds equals minus the Lie algebra bracket of their generators. 13.5 Reduction and Reconstruction of Dynamics Reduction of Dynamics. In the last sections we have focussed on reducing the Poisson structure from T ∗ G to g∗ . However, it is also very important to reduce the dynamics of a given Hamiltonian. The next theorem treats this, which is very useful in examples. Theorem 13.5.1 (Lie–Poisson Reduction of Dynamics). Let G be a Lie group and H : T ∗ G → R. Assume H is left (respectively, right) invariant. Then the function H − := H|g∗ (respectively, H + := H|g∗ ) on g∗ satisﬁes H = H − ◦ JR , that is, H(αg ) = H − (JR (αg )) ∗ for all αg ∈ Tg G (13.5.1) where JR : T ∗ G → g∗ is given by JR (αg ) = T ∗ Lg · αg (respectively, H = − H + ◦ JL , that is, ∗ H(αg ) = H + (JL (αg )) for all αg ∈ Tg G, (13.5.2) ........................... 15 July 1998—18h02 ...........................

448. 434 13. Lie–Poisson and Euler–Poincar´ Reduction e where JL : T ∗ G → g∗ is given by JL (αg ) = T ∗ Rg · αg ). + The flow Ft of XH on T ∗ G and the flow Ft− (respectively, Ft+ ) of XH − (respectively, XH + ) on g∗ (respectively, g∗ ) are related by − + JR (Ft (αg )) = Ft− (JR (αg )), (13.5.3) JL (Ft (αg )) = Ft+ (JL (αg )). (13.5.4) In other words, a left invariant Hamiltonian on T ∗ G induces Lie–Poisson dynamics on g∗ , while a right invariant one induces Lie–Poisson dynamics − on g∗ . The result is a direct consequence of the Lie–Poisson reduction + theorem and the fact that a Poisson map relates Hamiltonian systems and their integral curves to Hamiltonian systems. Left and Right Reductions. Above we saw that left reduction is im- plemented by the right momentum map. That is, H and H − as well as XH and XH − are JR -related if H is left invariant. We can get additional information using the fact that JL is conserved. Proposition 13.5.2. Let H : Te G be left invariant and H − be its re- ∗ ∗ ∗ striction to g as above. Let α(t) ∈ Tg(t) G be an integral curve of XH and let µ(t) = JR (α(t)) and ν(t) = JL (α(t)) so that ν is constant in time. Then ν = g(t) · µ(t) := Ad∗ −1 µ(t). g(t) (13.5.5) ∗ ∗ Proof. This follows from ν = Te Rg(t) α(t), µ(t) = Te Lg(t) α(t), the definition of the coadjoint action, and the fact that JL is conserved. Equation (13.5.5) already determines g(t) in terms of ν and µ(t) to some extent; for example, for SO(3) it says that g(t) rotates the vector µ(t) to the fixed vector ν. The Reconstruction Equation. Differentiating (13.5.5) in t and using the formulas for differentiating curves from §9.3, we get dµ 0 = g(t) · ξ(t) · µ(t) + , dt where ξ(t) = g(t)−1 g(t) and ξ · µ = − ad∗ µ. ˙ ξ On the other hand, µ(t) satisfies the Lie-Poisson equations ˙ dµ = ad∗ − /δµ µ, δH dt and so ξ(t) · µ(t) + ad∗ − /δµ µ(t) = 0; δH that is, ad∗ (−ξ(t)+δH − /δµ) µ(t) = 0. ........................... 15 July 1998—18h02 ...........................

449. 13.5 Reduction and Reconstruction of Dynamics 435 A sufficient condition for this is that ξ(t) = δH − /δµ; that is, δH − g(t)−1 g(t) = ˙ , (13.5.6) δµ which is called the reconstuction equation. Thus, it is plausible that we can reconsturct α(t) from µ(t) by first solving (13.5.6) with appropriate initial conditions and then letting ∗ α(t) = Tg(t) Lg(t)−1 µ(t). (13.5.7) This gives us a way to go back and forth between T ∗ G and g∗ : Lie–Poisson reduction ∗ E ∗ T G ' g . reconstruction We now look at the reconstruction procedure a little more closely and from a slightly different point of view. Left Trivialization of Dynamics. The next propoistion describes the vector field XH in the left trivialization of T ∗ G as G × g∗ . Let λ : T ∗ G −→ G × g∗ be the diffeomorphism defined by ∗ λ(αg ) = (g, Te Lg (αg )) = (g, JR (αg )). (13.5.8) It is easily verified that λ is equivariant relative to the cotangent lift of left translations on G and the G-action on G × g∗ given by g · (h, µ) = Λg (h, µ) = (gh, µ), (13.5.9) ∗ ∗ where g, h ∈ G and µ ∈ g . Let p1 : G × g → G denote the projection to the first factor. Note that p1 ◦ λ = π, where π : T ∗ G → G is the canonical cotangent bundle projection. Proposition 13.5.3. For g ∈ G, µ ∈ g∗ , the push-forward of XH by λ to G × g∗ is the vector field given by δH − (λ∗ XH )(g, µ) = Te Lg , µ, ad∗ − /δµ µ δH ∈ Tg G × Tµ g∗ , (13.5.10) δµ ........................... 15 July 1998—18h02 ...........................

450. 436 13. Lie–Poisson and Euler–Poincar´ Reduction e where H − = H|g∗ . Proof. As we have already shown, the map JR : T ∗ G −→ g∗ can be regarded as the standard projection to the quotient T ∗ G −→ T ∗ G/G for the left action, so that the second component of λ∗ XH is the Lie–Poisson reduction of XH and hence equals the Hamiltonian vector field XH − on g∗ . By Proposition 10.7.1 we can conclude that − (λ∗ XH )(g, µ) = (X µ (g), µ, ad∗ − /δµ µ), δH (13.5.11) where X µ ∈ X(G) is a vector field on G depending smoothly on the parameter µ ∈ g∗ . Since H is left invariant, so is XH and, by equivariance of the diffeomorphism λ, we also have Λ∗ λ∗ XH = λ∗ XH for any g ∈ G. This, in turn, is g equivalent to Tgh Lg−1 X µ (gh) = X µ (h) for all g, h ∈ G, and µ ∈ g∗ ; that is, X µ (g) = Te Lg X µ (e), (13.5.12) In view of (13.5.11) and (13.5.12), the proposition is proved if we show that δH − X µ (e) = . (13.5.13) δµ To prove this, we begin by noting that X µ (e) = T(e,µ) p1 (λ∗ XH (µ)) = (T(e,µ) p1 ◦ Tµ λ)XH (µ) = Tµ (p1 ◦ λ)XH (µ) = Tµ π(XH (µ)). (13.5.14) For a fixed ν ∈ g∗ , introduce the flow ∗ Ftν (αg ) = αg + tTe Lg (ν), (13.5.15) which leaves the fibers of T ∗ G invariant and therefore defines a vertical vector field Vν on T ∗ G (that is, T π ◦ Vν = 0) given by d ∗ Vν (αg ) = (αg + tTe Lg (ν)). (13.5.16) dt t=0 The defining relation iXH Ω = dH of XH evaluated at µ in the direction Vν (µ) gives Ω(µ)(XH (µ), Vν (µ)) = dH(µ) · Vν (µ) d δH − = H(µ + tν) = ν, . dt t=o δµ ........................... 15 July 1998—18h02 ...........................

451. 13.5 Reduction and Reconstruction of Dynamics 437 so that using Ω = −dΘ, we get dH − −XH [Θ(Vν )](µ) + Vν [Θ(XH )](µ) + Θ([XH , Vν ])(µ) = ν, . δµ (13.5.17) We will compute each term on the left-hand side of (13.5.17). Since Vν is vertical, T π ◦ Vν = 0, and so, by the defining formula for the canonical one-form, Θ(Vν ) = 0. The first term thus vanishes. To compute the second term, we use the definition of Θ and (13.5.14) to get d Vν [Θ(XH )](µ) = Θ(XH )(µ + tν) dt t=0 d = µ + tν, Tµ+tν π (XH (µ + tν)) dt t=0 d = µ + tν, X µ+tν (e) dt t=0 d = ν, X µ (e) + µ, X µ+tν (e) . (13.5.18) dt t=0 Finally, to compute the third term, we again use the defintion of Θ, the linearity of Tµ π to interchange the order of Tµ π and d/dt, the relation π ◦ Ftν = π, and (13.5.14) to get Θ([XH , Vν ])(µ) = µ, Tµ π · [XH , Vν ](µ) d = − µ, Tµ π · ((Ftν )∗ XH )(µ) dt t=0 d = − µ, Tµ π · Tµ+tν F−t (XH (µ + tν)) ν dt t=0 d = − µ, Tµ+tν (π ◦ F−t )(XH (µ + tν)) ν dt t=0 d = − µ, Tµ+tν π · XH (µ + tν) dt t=0 d = − µ, X µ+tν (e) . (13.5.19) dt t=0 Adding (13.5.18) and (13.5.19), and using (13.5.17) gives δH − ν, X µ (e) = ν, , δµ and thus (13.5.13) follows, thereby proving the proposition. ........................... 15 July 1998—18h02 ...........................

452. 438 13. Lie–Poisson and Euler–Poincar´ Reduction e The Reconstruction Theorem. This result now follows from what we have done. Theorem 13.5.4 (Lie-Poisson Reconstruction of Dynamics). Let G be a Lie group and H : T ∗ G → R be a left invariant Hamiltonian. Let H − = H|g∗ and let µ(t) be the integral curve of the Lie–Poisson equations. dµ = ad∗ − /δµ µ δH (13.5.20) dt ∗ with initial condition µ(0) = Te Lg0 (αg0 ). Then the integral curve α(t) ∈ ∗ Tg(t) G of XH with initial condition α(0) = αg0 is given by ∗ α(t) = Tg(t) Lg(t)−1 µ(t), (13.5.21) where g(t) is the solution of the equation g −1 g = δH − /δµ, that is, ˙ dg(t) δH − = Te Lg(t) , (13.5.22) dt δµ(t) with initial condition g(0) = g0 . Proof. The curve α(t) is the unique integral curve of XH with initial condition α(0) = αg0 if and only if ∗ λ(α(t)) = (g(t), Te Lg(t) α(t)) = (g(t), JR (α(t))) =: (g(t), µ(t)) is the integral curve of λ∗ XH with initial condition ∗ λ(α(0)) = (g0 , Te Lg0 (αg0 )) , which is equivalent to the statement in the theorem in view of (13.5.10). For right invariant Hamiltonians H : T ∗ G → R, we let H + = H|g∗ , the Lie–Poison equations are dµ = − ad∗ + /δµ µ, δH (13.5.23) dt the reconstruction formula is ∗ α(t) = Tg(t) Rg(t)−1 µ(t), (13.5.24) and the equation that g(t) satisﬁes is gg −1 = δH + /δµ, that is, ˙ dg(t) δH + = Te Rg(t) ; (13.5.25) dt δµ(t) the initial conditons remain unchanged. ........................... 15 July 1998—18h02 ...........................

453. 13.5 Reduction and Reconstruction of Dynamics 439 Lie–Poisson Reconstruction and Lagrangians. It is useful to keep in mind that the Hamiltonian H on T ∗ G often arises from a Lagrangian L : T G → R via a Legendre transform FL. In fact, many of the constructions and verifications are simpler using the Lagrangian formalism. Assume that L is left invariant (respectively, right invariant); that is, L(T Lg · v) = L(v), (13.5.26) respectively, L(T Rg · v) = L(v) (13.5.27) for all g ∈ G and v ∈ Th G. Differentiating (13.5.26) and (13.5.27), we find FL(T Lg · v) · (T Lg · w) = FL(v) · w, (13.5.28) respectively, FL(T Rg · v) · (T Rg · w) = FL(v) · w (13.5.29) for all v, w ∈ Th G and g ∈ G. In other words, T ∗ Lg ◦ FL ◦ T Lg = FL, (13.5.30) respectively, T ∗Rg ◦ FL ◦ T Rg = FL. (13.5.31) Note that the action of L is also left (respectively, right) invariant A(T Lg · v) = A(v), (13.5.32) respectively, A(T Rg · v) = A(v) (13.5.33) since A(T Lg · v) = FL(T Lg · v) · (T Lg · v) = FL(v) · v = A(v) by (13.5.28). Thus, the energy E = A − L is left (respectively, right) invariant on T G. If L is hyperregular, so FL : T G → T ∗ G is a diffeomorphism, then H = E ◦ (FL)−1 is left (respectively, right) invariant on T ∗ G. Theorem 13.5.5 (Alternative Lie–Poisson Reconstruction). Let L : T G → R be a hyperregular Lagrangian which is left (respectively, right) invariant on T G. Let H : T ∗ G → R be the associated Hamiltonian and H − : g∗ → R (respectively, H + : g∗ → R) be the induced Hamiltonian − + on g∗ . Let µ(t) ∈ g∗ be an integral curve for H − (respectively, H + ) with ........................... 15 July 1998—18h02 ...........................

454. 440 13. Lie–Poisson and Euler–Poincar´ Reduction e ∗ ∗ initial condition µ(0) = Te Lg0 · αg0 (respectively, µ(0) = Te Rg0 · αg0 ) and −1 let ξ(t) = FL µ(t) ∈ g. Let v0 = Te Lg0 · ξ(0) ∈ Tg0 G. Then the integral curve for the Lagrangian vector field associated with L with initial condition (g0 , v0 ) is given by VL (t) = Te Lg(t) · ξ(t), (13.5.34) respectively, VR (t) = Te Rg (t) · ξ(t), (13.5.35) where g(t) solves the equation g −1 g = ξ; that is, ˙ dg = Te Lg(t) · ξ(t), g(0) = g0 , (13.5.36) dt respectively, g −1 g = ξ that is, ˙ dg = Te Rg(t) · ξ(t), g(0) = g0 . (13.5.37) dt The corresponding integral curve of XH on T ∗ G with initial condition αg0 and covering µ(t) is ∗ α(t) = FL(VL (t)) = Tg(t) L(g(t))−1 µ(t), (13.5.38) respectively, ∗ α(t) = FL(VR (t)) = Tg(t) R(g(t))−1 µ(t). (13.5.39) Proof. This follows from Theorem 13.5.5 by applying FL−1 to (13.5.21) and (13.5.24) respectively. As for the equation satisfied by g(t), since the Lagrangian vector field XE is a second-order equation, we necessarily have dg = VL (t) = Te Lg(t) ξ(t) dt and dg = VR (t) = Te Rg(t) ξ(t), dt respectively. Thus, given ξ(t), one solves (13.5.36) for g(t) and then constructs VL (t) or α(t) from (13.5.34) and (13.5.38). As we shall see in the examples, this procedure has a natural physical interpretation. The previous theorem generalizes to arbitrary Lagrangian systems in the following way. In fact, The- orem 13.5.5 is a corollary of the next theorem. ........................... 15 July 1998—18h02 ...........................

455. 13.5 Reduction and Reconstruction of Dynamics 441 Theorem 13.5.6 (Lagrangian Lie–Poisson Reconstruction). Let L : TG → R be a left invariant Lagrangian such that its Lagrangian vector field Z ∈ X(T G) is a second-order equation and is left invariant. Let ZG ∈ X(g) be the induced vector field on (T G)/G ≈ g and let ξ(t) be an integral curve of ZG . If g(t) ∈ G is the solution of the nonautonomous ordinary differential equation g(t) = Te Lg(t) ξ(t), g(0) = e, g ∈ G, ˙ then V (t) = Te Lgg(t) ξ(t) is the integral curve of Z satisfying V (0) = Te Lg ξ(0) and V (t) projects to ξ(t), that is, T Lτ (V (t))−1 V (t) = ξ(t), where τ : T G → G is the tangent bundle projection. Proof. Let V (t) be the integral curve of Z satisfying V (0) = Te Lg ξ(0) for a given element ξ(0) ∈ g. Since ξ(t) is the integral curve of ZG whose flow is conjugated to the flow of Z by left translation, we have T Lτ (V (t))−1 V (t) = ξ(t). If h(t) = τ (V (t)), since Z is a second-order equation, we have ˙ V (t) = h(t) = Te Lh(t) ξ(t), h(0) = τ (V (0)) = g, so that, letting g(t) = g −1 h(t) we get g(0) = e and ˙ g(t) = T Lg−1 h(t) = T Lg−1 T Lh(t) ξ(t) = T Lg(t) ξ(t). ˙ This determines g(t) uniquely from ξ(t) and so V (t) = Te Lh(t) ξ(t) = Te Lgg(t) ξ(t). These calculations suggest rather strongly that one should examine the Lagrangian (rather than the Hamiltonian) side of the story on an independent footing. We will do exactly that shortly. ........................... 15 July 1998—18h02 ...........................

456. 442 13. Lie–Poisson and Euler–Poincar´ Reduction e The Lie–Poisson–Hamilton–Jacobi Equation. Since Poisson brackets and Hamilton’s equations naturally drop from T ∗ G to g∗ , it is natural to ask if other structures do too, such as Hamilton–Jacobi theory. We investigate this question now, leaving the proofs and related remarks to the internet supplement. Add proof Let H be a G invariant function on T ∗ G and let H − be the corresponding to internet left reduced Hamiltonian on g∗ . (To be specific, we deal with left actions; of supplement! course, there are similar statements for right reduced Hamiltonians). If S is invariant, there is a unique function S − such that S(g, g0 ) = S − (g −1 g0 ). −1 (One gets a slightly different representation for S by writing g0 g in place −1 of g g0 .) Proposition 13.5.7 (Ge Marsden [1988]). The left reduced Ham- ilton–Jacobi equation is the following equation for a function S − : G → R: ∂S − + H − (−T Rg · dS − (g)) = 0, ∗ (13.5.40) ∂t which is called the Lie–Poisson Hamilton–Jacobi equation. The Lie– Poisson flow of the Hamiltonian H − is generated by the solution S − of (13.5.40) in the sense that the flow is given by the Poisson transformation Π0 → Π of g∗ defined as follows. Define g ∈ G by solving the equation Π0 = −T L∗ · dg S − g (13.5.41) for g ∈ G and then set Π = g · Π0 = Ad∗−1 Π0 . g (13.5.42) The action in (13.5.42) is the coadjoint action. Note that (13.5.42) and (13.5.41) give Π = −T Rg · dS − (g). ∗ Exercises 13.5-1. Write out the reconstruction equations for the group G = SO(3). 13.5-2. Write out the reconstruction equations for the group G = Diff vol (Ω). 13.5-3. Write out the Lie–Poisson Hamilton–Jacobi equation for SO(3). 13.6 The Linearized Lie–Poisson Equations Here we show that the equations linearized about an equilibrium solution of a Lie–Poisson system (such as the ideal fluid equations) are Hamiltonian with respect to a “constant coefficient” Lie–Poisson bracket. The Hamil- tonian for these linearized equations is 1 δ 2 (H + C)|e , the quadratic func- 2 tional obtained by taking one-half of the second variation of the Hamilto- nian plus conserved quantities and evaluating it at the equilibrium solution ........................... 15 July 1998—18h02 ...........................

457. 13.6 The Linearized Lie–Poisson Equations 443 where the conserved quantity C (often a Casimir) is chosen so that the first variation δ(H + C) vanishes at the equilibrium. A consequence is that the linearized dynamics preserves 1 δ 2 (H + C)|e . This is useful for studying 2 stability of the linearized equations. For a Lie algebra g, recall that the Lie–Poisson bracket is defined on g∗ , the dual of g with respect to (a weakly nondegenerate) pairing , between g∗ and g by the usual formula δF δG {F, G} (µ) = µ, , , (13.6.1) δµ δµ where δF/δµ ∈ g is determined by δF DF (µ) · δµ = δµ, (13.6.2) δµ when such an element δF/δµ exists, for any µ, δµ ∈ g∗ . The equations of motion are ∗ dµ δH = − ad µ, (13.6.3) dt δµ where H : g∗ → R is the Hamiltonian, ad(ξ) : g → g is the adjoint action, ad(ξ)·η = [ξ, η] for ξ, η ∈ g, and ad(ξ)∗ : g∗ → g∗ is its dual. Let µe ∈ g∗ be an equilibrium solution of (13.6.3). The linearized equations of (13.6.3) at µe are obtained by expanding in a Taylor expansion with small parameter ε using µ = µe + εδµ, and taking (d/dε)|ε=0 of the resulting equations. This gives δH δH δH = + εD (µe ) · δµ + O(ε2 ), (13.6.4) δµ δµe δµ where δH/δµe , δµ := DH(µe ) · δµ, and the derivative D(δH/δµ)(µe ) · δµ is the linear functional ν ∈ g∗ → D2 H(µe ) · (δµ, ν) ∈ R (13.6.5) by using the definition (13.6.2). Since δ 2 H(δµ) := D2 H(µe ) · (δµ, δµ), it follows that the functional (13.6.5) equals 1 δ(δ 2 H) . 2 δ(δµ) Consequently, (13.6.4) becomes δH δH 1 δ(δ 2 H) = + ε + O(ε2 ) (13.6.6) δµ δµe 2 δ(δµ) ........................... 15 July 1998—18h02 ...........................

458. 444 13. Lie–Poisson and Euler–Poincar´ Reduction e and the Lie–Poisson equations (13.6.3) yield ∗ dµe d(δµ) δH +ε = − ad µe dt dt δµe ∗ ∗ 1 δ(δ 2 H) δH − ε ad µe − ad δµ + O(ε2 ). 2 δ(δµ) δµe Thus, the linearized equations are ∗ ∗ d(δµ) 1 δ(δ 2 H) δH = − ad µe − ad δµ. (13.6.7) dt 2 δ(δµ) δµe If H is replaced by HC := H + C, with the Casimir function C chosen to satisfy δHC /δµe = 0, we get ad(δHC /δµe )∗ δµ = 0, and so ∗ d(δµ) 1 δ(δ 2 HC ) = − ad µe . (13.6.8) dt 2 δ(δµ) Equation (13.6.8) is Hamiltonian with respect to the linearized Poisson bracket (see Example (f) of §10.1): δF δG {F, G} (µ) = µe , , . (13.6.9) δµ δµ Ratiu [1982] interprets this bracket in terms of a Lie–Poisson structure of a loop extension of g. The Poisson bracket (13.6.9) differs from the Lie– Poisson bracket (13.6.1) in that it is constant in µ. With respect to the Poisson bracket (13.6.9), Hamilton’s equations given by δ 2 HC are (13.6.8), as an easy verification shows. Note that the critical points of δ 2 HC are stationary solutions of the linearized equation (13.6.8), that is, they are neutral modes for (13.6.8). If δ 2 HC is definite, then either δ 2 HC or −δ 2 HC is positive-definite and hence defines a norm on the space of perturbations δµ (which is g∗ ). Being twice the Hamiltonian function for (13.6.8), δ 2 HC is conserved. So, any solution of (13.6.8) starting on an energy surface of δ 2 HC (i.e., on a sphere in this norm) stays on it and hence the zero solution of (13.6.8) is (Liapunov) stable. Thus, formal stability, i.e., definiteness of δ 2 HC , implies linearized stability. It should be noted, however, that the conditions for definiteness of δ 2 HC are entirely different from the conditions for “normal mode stability,” that is, that the operator acting on δµ given by (13.6.8) have a purely imaginary spectrum. In particular, having a purely imaginary spectrum for the linearized equation does not produce Liapunov stability of the linearized equations. The difference between δ 2 HC and the operator in (13.6.8) can be made explicit, as follows. Assume that there is a weak Ad-invariant metric , on g and a linear operator L : g → g such that δ 2 HC = δµ, Lδµ ; (13.6.10) ........................... 15 July 1998—18h02 ...........................

459. 13.7 The Euler–Poincar´ Equations e 445 L is symmetric with respect to the metric , , that is, ξ, Lη = Lξ, η for all ξ, η ∈ g. Then the linear operator in (13.6.8) becomes δµ → [Lδµ, µe ] (13.6.11) which, of course, differs from L, in general. However, note that the kernel of L is included in the kernel of the linear operator (13.6.11), that is, the zero eigenvalues of L give rise to “neutral modes” in the spectral analysis of (13.6.11). There is a remarkable coincidence of the zero-eigenvalue equations for these operators in fluid mechanics: for the Rayleigh equation describing plane-parallel shear flow in an inviscid homogeneous fluid, taking normal modes makes the zero-eigenvalue equations corresponding to L and to (13.6.11) coincide (see Abarbanel, Holm, Marsden, and Ratiu [1986]). For additional applications of the stability method, see the Introduction and Holm, Marsden, Ratiu, and Weinstein [1985], Abarbanel and Holm [1987], Simo, Posbergh, and Marsden [1990, 1991], and Simo, Lewis, and Marsden [1991]. For a more general treatment of the linearization process, see Marsden, Ratiu, and Raugel [1991]. Exercises 13.6-1. Write out the linearized rigid body equations about an equilibrium explicitly. 13.6-2. Let g be finite dimensional. Let e1 , . . . , en be a basis for g and e1 , . . . , en a dual basis for g∗ . Let µ = µa ea ∈ g∗ and H(µ) = H(µ1 , . . . , µn ) : g∗ → R. Let [µa , µb ] = Cab µd . Derive a coordinate expression for the lin- d earized equations (13.6.7): ∗ ∗ d(δµ) 1 δ(δ 2 H) δH = − ad µe − ad δµ. dt 2 δµ δµe 13.7 The Euler–Poincar´ Equations e Some History of Lie–Poisson and Euler–Poincar´ Equations. We e continue with some comments on the history of Poisson structures that we began in §10.3. Recall that we pointed out how Lie, in his work up to 1890 on function groups, had many of the essential ideas of general Poisson manifolds and, in particular, had explicitly studied the Lie–Poisson bracket on duals of Lie algebras. The theory developed so far in this chapter describes the adaptation of the concepts of Hamiltonian mechanics to the context of the duals of Lie algebras. This theory could easily have been given shortly after Lie’s work, but evidently it was not observed for the rigid body or ideal fluids ........................... 15 July 1998—18h02 ...........................

460. 446 13. Lie–Poisson and Euler–Poincar´ Reduction e until the work of Pauli [1953], Martin [1959], Arnold [1966a], Ebin and Marsden [1970], Nambu [1973], and Sudarshan and Mukunda [1974], all of whom were apparently unaware of Lie’s work on the Lie–Poisson bracket. It seems that even Elie Cartan was unaware of this aspect of Lie’s work, which does seem surprising. Perhaps it is less surprising when one thinks for a moment about how many other things Cartan was involved in at the time. Nevertheless, one is struck by the amount of rediscovery and confusion in this subject. Evidently, this situation is not unique to mechanics. Meanwhile, as Arnold [1988] and Chetaev [1989] pointed out, one can also write the equations directly on the Lie algebra, bypassing the Lie– Poisson equations on the dual. The resulting equations were first written down on a general Lie algebra by Poincar´ [1901b]; we refer to these as the e Euler–Poincar´ equations. We shall develop them from a modern point of e view in the next section. Poincar´ [1910] goes on to study the effects of e the deformation of the earth on its precession—he apparently recognizes the equations as Euler equations on a semidirect product Lie algebra. In general, the command that Poincar´ had of the subject is most impressive, e and is hard to match in his near contemporaries, except perhaps Riemann [1860, 1861] and Routh [1877, 1884]. It is noteworthy that Poincar´ [1901b] e has no references, so it is rather hard to trace his train of thought or his sources; compare this style with that of Hamel [1904]! In particular, he gives no hint that he understood the work of Lie on the Lie–Poisson structure, but, of course, Poincar´ understood the Lie group and the Lie e algebra machine very well indeed. Our derivation of the Euler–Poincar´ equations in the next section is e based on a reduction of variational principles, not on a reduction of the symplectic or Poisson structure, which is natural for the dual. We also show that the Lie–Poisson equations are related to the Euler–Poincar´ equations e by the “fiber derivative,” in the same way as one gets from the ordinary Euler–Lagrange equations to the Hamilton equations. Even though this is relatively trivial, it does not appear to have been written down before. In the dynamics of ideal fluids, the resulting variational principle is related to what has been known as “Lin constraints” (see also Newcomb [1962] and Bretherton [1970].) This itself has an interesting history, going back to Ehrenfest, Boltzman, and Clebsch, but again, there was little if any contact with the heritage of Lie and Poincar´ on the subject. One person who was e well aware of the work of both Lie and Poincar´ was Hamel. e How does Lagrange fit into this story? In Mecanique Analytique, Vol- ume 2, equations A on page 212 are the Euler–Poincar´ equations for the e rotation group written out explicitly for a reasonably general Lagrangian. He eventually specializes them to the rigid body equations of course. We should remember that Lagrange also developed the key concept of the La- grangian representation of fluid motion, but it is not clear that he understood that both systems are special instances of one theory. Lagrange spends a large number of pages on his derivation of the Euler–Poincar´ e ........................... 15 July 1998—18h02 ...........................

461. 13.7 The Euler–Poincar´ Equations e 447 equations for SO(3), in fact, a good chunk of Volume 2. His derivation is not as clean as we would give today, but it seems to have the right spirit of a reduction method. That is, he tries to get the equations from the Euler–Lagrange equations on T SO(3) by passing to the Lie algebra. In view of the historical situation described above, one might argue that the term “Euler–Lagrange-Poincar´” equations is right for these equations. e Since Poincar´ noted the generalization to arbitrary Lie algebras, and ap- e plied it to interesting fluid problems, it is clear that his name belongs, but in light of other uses of the term “Euler–Lagrange,” it seems that “Euler– Poincar´” is a reasonable choice. e Marsden and Scheurle [1993a,b] and Weinstein [1994] have studied a more general version of Lagrangian reduction whereby one drops the Euler– Lagrange equations from T Q to T Q/G. This is a nonabelian generalization of the classical Routh method, and leads to a very interesting coupling of the Euler–Lagrange and Euler–Poincar´ equations that we shall briefly e sketch in the next section. This problem was also studied by Hamel [1904] in connection with his work on nonholonomic systems (see Koiller [1992] and Bloch, Krishnaprasad, Marsden, and Murray [1994] for more information). The current vitality of mechanics, including the investigation of fundamental questions, is quite remarkable, given its long history and development. This vitality comes about through rich interactions with both pure mathematics (from topology and geometry to group representation theory), and through new and exciting applications to areas like control theory. It is perhaps even more remarkable that absolutely fundamental points, such as a clear and unambiguous linking of Lie’s work on the Lie–Poisson bracket on the dual of a Lie algebra and Poincar´’s work on the Euler–Poincar´ e e equations on the Lie algebra itself, with the most basic of examples in mechanics, such as the rigid body and the motion of ideal fluids, took nearly a century to complete. The attendant lessons to be learned about commu- nication between pure mathematics and the other mathematical sciences are, hopefully, obvious. Rigid Body Dynamics. To understand this section, it will be helpful to develop some more of the basics about rigid body dynamics from the Introduction (further details are given in Chaper 15). We regard an element R ∈ SO(3) giving the configuration of the body as a map of a reference configuration B ⊂ R3 to the current configuration R(B); the map R takes a reference or label point X ∈ B to a current point x = RX ∈ R(B). See Figure ??. When the rigid body is in motion, the matrix R is time-dependent and the velocity of a point of the body is x = RX = RR−1 x. Since R is an ˙ ˙ ˙ −1 ˙ ˙ −1 are skew matrices, and so we can orthogonal matrix, R R and RR write x = RR−1 x = ω × x, ˙ ˙ (13.7.1) ........................... 15 July 1998—18h02 ...........................

462. 448 13. Lie–Poisson and Euler–Poincar´ Reduction e R X R(B) x B reference configuration current configuration Figure 13.7.1. The rotation R takes the reference configuration to the current configuration. which defines the spatial angular velocity vector ω. Thus, ω is given ˙ by right translation of R to the identity. The corresponding body angular velocity is defined by Ω = R−1 ω, (13.7.2) so that Ω is the angular velocity relative to a body fixed frame. Notice that R−1 RX = R−1 RR−1 x = R−1 (ω × x) ˙ ˙ = R−1 ω × R−1 x = Ω × X (13.7.3) ˙ so that Ω is given by left translations of R to the identity. The kinetic 2 energy is obtained by summing up m x /2 over the body: ˙ K= 1 ˙ ρ(X) RX 2 d3 X, (13.7.4) 2 B where ρ is a given mass density in the reference configuration. Since RX = ω × x = R−1 (ω × x) = Ω × X , ˙ K is a quadratic function of Ω. Writing K = 1 ΩT IΩ 2 (13.7.5) defines the moment of inertia tensor I, which, if the body does not degenerate to a line, is a positive-definite (3×3)-matrix, or better, a quadratic form. This quadratic form can be diagonalized, and this defines the principal axes and moments of inertia. In this basis, we write I = diag(I1 , I2 , I3 ). The function K is taken to be the Lagrangian of the system on T SO(3) (and by means of the Legendre transformation we get the corresponding Hamiltonian description on T ∗ SO(3)). Notice directly from (13.7.4) that K is left (not right) invariant on T SO(3). It follows that the corresponding Hamiltonian is also left invariant. ........................... 15 July 1998—18h02 ...........................

463. 13.7 The Euler–Poincar´ Equations e 449 Dynamics in the Group vs. the Algebra. From the Lagrangian point of view, the relation between the motion in R space and that in body angular velocity (or Ω) space is as follows: Theorem 13.7.1. The curve R(t) ∈ SO(3) satisfies the Euler–Lagrange equations for the Lagrangian ˙ L(R, R) = 1 ˙ ρ(X) RX 2 d3 X, (13.7.6) 2 B if and only if Ω(t) defined by R−1 Rv = Ω × v for all v ∈ R3 satisfies ˙ Euler’s equations ˙ IΩ = IΩ × Ω. (13.7.7) One instructive way to prove this indirectly is to pass to the Hamiltonian formulation and to use Lie–Poisson reduction. One way to do it directly is to use variational principles. By Hamilton’s principle, R(t) satisfies the Euler–Lagrange equations if and only if δ L dt = 0. ˙ Let l(Ω) = 1 (IΩ) · Ω, so that l(Ω) = L(R, R) if R and Ω are related as 2 above. To see how we should transform Hamilton’s principle, we differentiate the relation R−1 R = Ω with respect to R to get ˙ ˆ −R−1 (δR)R−1 R + R−1 (δ R) = δΩ. ˙ ˙ (13.7.8) ˆ Let the skew matrix Σ be defined by Σ = R−1 δR ˆ (13.7.9) and define the corresponding vector Σ, as usual, by ˆ Σv = Σ × v. (13.7.10) Note that ˙ Σ = −R−1 RR−1 δR + R−1 δ R, ˆ ˙ ˙ so ˙ R−1 δ R = Σ + R−1 RΣ. ˙ ˆ ˙ ˆ (13.7.11) Substituting (13.7.11) and (13.7.9) into (13.7.8) gives ˆˆ ˙ ˆ ˆˆ −ΣΩ + Σ + ΩΣ = δΩ, ........................... 15 July 1998—18h02 ...........................

464. 450 13. Lie–Poisson and Euler–Poincar´ Reduction e that is, ˙ ˆ ˆ ˆ δΩ = Σ + [Ω, Σ]. (13.7.12) ˆ ˆ The identity [Ω, Σ] = (Ω × Σ) holds by Jacobi’s identity for the cross ˆ product, and so ˙ δΩ = Σ + Ω × Σ. (13.7.13) These calculations prove the following: Theorem 13.7.2. Hamilton’s variational principle b δ L dt = 0 (13.7.14) a on T SO(3) is equivalent to the reduced variational principle b δ l dt = 0 (13.7.15) a on R3 where the variations δΩ are of the form (13.7.13) with Σ(a) = Σ(b) = 0. Proof of Theorem 13.7.1. It suﬃces to work out the equations equivalent to the reduced variational principle (13.7.15). Since l(Ω) = IΩ, Ω /2, and I is symmetric, we get b b δ l dt = IΩ, δΩ dt a a b = ˙ IΩ, Σ + Ω × Σ dt a b d = − IΩ, Σ + IΩ, Ω × Σ a dt b d = − IΩ + IΩ × Ω, Σ dt, a dt where we have integrated by parts and used the boundary conditions Σ(b) = Σ(a) = 0. Since Σ is otherwise arbitrary, (13.7.15) is equivalent to d − (IΩ) + IΩ × Ω = 0, dt which are Euler’s equations. ........................... 15 July 1998—18h02 ...........................

465. 13.7 The Euler–Poincar´ Equations e 451 Euler–Poincar´ Reduction. We now generalize this procedure to an e arbitrary Lie group and later will make the direct link with the Lie–Poisson equations. Theorem 13.7.3. Let G be a Lie group and let L : T G → R be a left invariant Lagrangian. Let l : g → R be its restriction to the identity. For a curve g(t) ∈ G, let ξ(t) = g(t)−1 · g(t); that is, ξ(t) = Tg(t) Lg(t)−1 g(t). ˙ ˙ Then the following are equivalent: (i) g(t) satisfies the Euler–Lagrange equations for L on G; (ii) the variational principle δ L(g(t), g(t)) dt = 0 ˙ (13.7.16) holds, for variations with fixed endpoints; (iii) the Euler–Poincar´ equations hold: e d δl δl = ad∗ ; ξ (13.7.17) dt δξ δξ (iv) the variational principle δ l(ξ(t)) dt = 0 (13.7.18) holds on g, using variations of the form δξ = η + [ξ, η], ˙ (13.7.19) where η vanishes at the endpoints. Proof. First of all, the equivalence of (i) and (ii) holds on the tangent bundle of any configuration manifold Q, as we know from Chapter 8. To see that (ii) and (iv) are equivalent, one needs to compute the variations δξ induced on ξ = g −1 g = T Lg−1 g by a variation of g. We will do this ˙ ˙ for matrix groups; see Bloch, Krishnaprasad, Marsden, and Ratiu [1994b] for the general case. To calculate this, we need to differentiate g −1 g in the ˙ direction of a variation δg. If δg = dg/d at = 0, where g is extended to a curve g , then, d d d2 g δξ = g −1 g = − g −1 δgg −1 g + g −1 ˙ , d dt =0 dt d =0 while if η = g −1 δg, then d d d2 g η= ˙ g −1 g = − g −1 gg −1 δg + g −1 ˙ . dt d =0 dt d =0 ........................... 15 July 1998—18h02 ...........................

466. 452 13. Lie–Poisson and Euler–Poincar´ Reduction e The difference δξ − η is thus the commutator [ξ, η]. ˙ To complete the proof, we show the equivalence of (iii) and (iv). Indeed, using the definitions and integrating by parts, δl δ l(ξ)dt = , δξ dt δξ δl = , (η + adξ η) dt ˙ δξ d δl δl = − + ad∗ ξ ,η dt dt δξ δξ so the result follows. There is of course a right invariant version of this theorem in which ξ = gg −1 and when (13.7.17), (13.7.19) acquire minus signs, that is, ˙ d δl δl = − ad∗ ξ and δξ = η − [ξ, η]. ˙ dt δξ δξ In coordinates, (13.7.17), reads as follows d ∂l ∂l = Cda ξ d b . b (13.7.20) dt ∂ξ a ∂ξ Euler–Poincar´ Reconstruction. On the Lagrangian side, reconstruc- e tion is very simple and centers on the reconstruction equation, which for left invariant systems reads g(t)−1 g(t) = ξ(t). ˙ (13.7.21) For the rigid body, this is just the definition of the body angular velocity Ω(t): R(t)−1 R(t) = Ω(t). ˙ ˆ (13.7.22) Reconstruction is read off Theorem 13.7.3 as follows. −1 Proposition 13.7.4. Let v0 ∈ Tg0 G, ξ0 = g0 v0 ∈ g and let ξ(t) be the solution of the Euler–Poincar´ equations with initial condition ξ0 . Solve the e reconstruction equation (13.7.21) for g(t) with g(0) = g0 . Then the solution of the Euler–Lagrange equations with initial condition v0 is v(t) ∈ Tg(t) G, given by v(t) = g(t) = g(t)ξ(t) ˙ (13.7.23) As mentioned earlier, to carry this out in examples, it is useful to make use of the conservation law to help solve the reconstruction equation. We shall see this in the case of the rigid body in Chapter 15. ........................... 15 July 1998—18h02 ...........................

467. 13.7 The Euler–Poincar´ Equations e 453 The Legendre Transformation. Since, in the hyperregular case, the Euler–Lagrange and Hamilton equations on T Q and T ∗ Q are equivalent, it follows that the Lie–Poisson and Euler–Poincar´ equations are also equiv- e alent. To see this directly, we make the following Legendre transformation from g to g∗ : δl µ= , h(µ) = µ, ξ − l(ξ). δξ Assuming the map ξ → µ is a diffeomorphism of g to g∗ , note that δh δξ δl δξ = ξ + µ, − , =ξ δµ δµ δξ δµ and so it is now clear that the Lie–Poisson and Euler–Poincar´ equations e are equivalent. The Virasoro Algebra. We close this section by showing that the periodic KdV equation, (see Example (c) in §3.2) ut + 6uux + uxxx = 0 is an Euler–Poincar´ equation on a certain Lie algebra called the Vira- e soro algebra v. These results were obtained in the Lie–Poisson context by Gelfand and Dorfman [1979], Kirillov [1981], Ovsienko and Khesin [1987], and Segal [1991]. See also Pressley and Segal [1986] and references therein. We begin with the construction of the Virasoro algebra v. If one identifies elements of X(S 1 ) with periodic functions of period 1 endowed with the Jacobi–Lie bracket [u, v] = uv − u v, the Gelfand–Fuchs cocycle is defined by the expression 1 Σ(u, v) = γ u (x)v (x)dx, 0 where γ ∈ R is a constant (to be determined later). The Lie algebra X(S 1 ) of vector fields on the circle has a unique central extension by R determined by the Gelfand-Fuchs cocycle. Therefore, (see (12.3.22) in Remark 5 of §12.3), the Lie algebra bracket on v := {(u, a) | u ∈ X(S 1 ), a ∈ R} is given by 1 [(u, a), (v, b)] = −uv + u v, γ u (x)v (x) dx 0 ........................... 15 July 1998—18h02 ...........................

468. 454 13. Lie–Poisson and Euler–Poincar´ Reduction e since the left Lie bracket on X(S 1 ) is given by the negative of the Jacobi– Lie bracket for vector fields. Identify the dual of v with v by the L2 -inner product 1 (u, a), (v, b) = ab + u(x)v(x) dx. 0 We claim that the coadjoint action ad∗ (u,a) is given by ad∗ (v, b) = (bγu + 2u v + uv , 0). (u,a) Indeed, if (u, a), (v, b), (w, c) ∈ v, we have ad∗ (v, b), (w, c) (u,a) = (v, b), [(u, a), (w, c)] 1 = (v, b), −uw + u w, γ u (x)w (x) dx 0 1 1 1 = bγ u (x)w (x) dx − v(x)u(x)w (x) dx + v(x)u (x)w(x) dx. 0 0 0 Integrating the first term twice and the second term once by parts and remembering that the boundary terms vanish by periodicity, this expresison becomes 1 1 1 bγ u (x)w(x)dx + (v(x)u(x)) w(x)dx + v(x)u (x)w(x)dx 0 0 0 1 = (bγu (x) + 2u (x)v(x) + u(x)v (x))w(x)dx 0 = (bγu + 2u v + uv , 0), (w, c) . The Euler–Poincar´ Form of the KdV Equation. If F : v → R, its e functional derivative relative to the L2 -pairing is given by δF δF ∂F = , δ(u, a) δu ∂a where δF/δu is the usual L2 -functional derivative of F keeping a ∈ R fixed and ∂F/∂a is the standard partial derivative of F keeping u fixed. The Euler–Poincar´ equations for right invariant systems with Lagrangian e l : v → R become d δl δl = − ad∗ (u,a) . dt δ(u, a) δ(u, a) ........................... 15 July 1998—18h02 ...........................

469. 13.7 The Euler–Poincar´ Equations e 455 However, δl δl ∂l ad∗ (u,a) = ad∗ (u,a) , δ(u, a) δu ∂a ∂l δl δl = γ u + 2u +u ,0 , ∂a δu δu so that the Euler–Poincar´ equations become the system e d ∂l =0 dt ∂a d δl ∂l δl δl = −γ u − 2u −u . dt δu ∂a δu δu If 1 1 l(u, a) = a2 + u2 (x) dx , 2 0 then ∂l/∂a = a, δl/δu = u and the above equations become da = 0 dt (13.7.24) du = −γau − 3u u. dt Since a is constant, we get ut + 3ux u + γau = 0. (13.7.25) This equation is equivalent to the KdV equation upon rescaling time and choosing the constant a appropriately. Indeed, let u(t, x) = v(τ (t), x) for τ (t) = t/2. Then ux = vx and ut = vτ /2 so that (13.7.25) can be written as vτ + 6vvx + 2γavxxx = 0, which becomes the KdV equation (see §3.2) if we choose a = 1/2γ. The Lie–Poisson form of the KdV equation. The (+) Lie–Poisson bracket is given by δ δh {f, h}(u, a) = (u, a), , δ(u, a) δ(u, a) δf δh δf δh = u − δu δu δu δu δf δh +aγ dx δu δu ........................... 15 July 1998—18h02 ...........................

470. 456 13. Lie–Poisson and Euler–Poincar´ Reduction e so that the Lie–Poisson equations f˙ = {f, h} become da = 0 dt (13.7.26) du δh δh δh = −u − 2u − aγ . dt δu δu δu Taking 1 1 2 1 h(u, a) = a + u2 (x) dx, 2 2 0 we get ∂h/∂a = a, δh/δu = u and so (13.7.26) becomes (13.7.25) as was to be expected and could have been directly obtained by a Legendre transform. The conclusion is that the KdV equation is the expression in space coordinates of the geodesic equations on the Virasoro group V endowed with the right invariant metric whose value at the identity is the L2 -inner product. We shall not describe here the Virasoro group which is a central extension of the diffeomorphism group on S 1 ; we refer the reader to Pressley and Segal [1986]. Exercises 13.7-1. Verify the coordinate form of the Euler–Poincar´ equations. e 13.7-2. Show that the Euler equations for a perfect fluid are Euler– Poincar´ equations. Find the variational principle (3) in Newcomb [1962] e and Bretherton [1970]. ˙ 13.7-3. Derive the rigid body Euler equations Π = Π × Ω directly from ˙ the momentum conservation law π = 0 and the relation π = RΠ. 13.8 The Lagrange–Poincar´ Equations e As we have mentioned, the Lie–Poisson and Euler–Poincar´ equations occur e for many systems besides the rigid body equations. They include the equations of fluid and plasma dynamics, for example. For many other systems, such as a rotating molecule or a spacecraft with movable internal parts, one has a combination of equations of Euler–Poincar´ type and Euler–Lagrange e type. Indeed, on the Hamiltonian side, this process has undergone development for quite some time. On the Lagrangian side, this process is also very interesting, and has been recently developed by, amongst others, Marsden and Scheurle [1993a,b], Holm, Marsden, and Ratiu [1998], and Cendra, Marsden, and Ratiu [1998]. In this section we just give a few indications of how this more general theory proceeds. ........................... 15 July 1998—18h02 ...........................

471. 13.8 The Lagrange–Poincar´ Equations e 457 The general problem is to drop Euler–Lagrange equations and variational principles from a general velocity phase-space T Q to the quotient T Q/G by a Lie group action of G on Q. If L is a G-invariant Lagrangian on T Q, it induces a reduced Lagrangian l on T Q/G. We give a brief preview of the general theory in this section. In fact, the material below can also act as motivation for the general theory of connections. An important ingredient in this work is to introduce a connection A on the principal bundle Q → S = Q/G, assuming that this quotient is nonsingular. For example, the mechanical connection (see Kummer [1981], Marsden [1992] and references therein), may be chosen for A. This connection allows one to split the variables into a horizontal and vertical part. Let xα , also called “internal variables,” be coordinates for shape-space Q/G, let η a be coordinates for the Lie algebra g relative to a chosen basis, let l be the Lagrangian regarded as a function of the variables xα , xα , η a , and ˙ a let Cdb be the structure constants of the Lie algebra g of G. If one writes the Euler–Lagrange equations on T Q in a local principal bundle trivialization, with coordinates xα on the base and η a in the ﬁber, then one gets the following system of Hamel equations: d ∂l ∂l α − α = 0, (13.8.1) dt ∂ x ˙ ∂x d ∂l ∂l a d − a Cdb η = 0. (13.8.2) dt ∂η b ∂η However, this representation of the equations does not make global intrinsic sense (unless Q → S admits a global ﬂat connection). The introduction of a connection overcomes this and one can intrinsically and globally split the original variational principle relative to horizontal and vertical variations. One gets from one-form to the other by means of the velocity shift given by replacing η by the vertical part relative to the connection ξ a = Aa xα + η a . α˙ Here, Ad are the local coordinates of the connection A. This change of α coordinates is motivated from the mechanical point of view since the variables ξ have the interpretation of the locked angular velocity. The resulting Lagrange–Poincar´ equations have the following form: e d ∂l ∂l ∂l α − α = a Bαβ xβ + Bαd ξ d , a ˙ a (13.8.3) dt ∂ x ˙ ∂x ∂ξ d ∂l ∂l = a (Bbα xα + Cdb ξ d ). a ˙ a (13.8.4) dt ∂ξ b ∂ξ a In these equations, Bαβ are the coordinates of the curvature B of A, Bdα = Cbd Ab a a α and Bbα = −Bαb . a a ........................... 15 July 1998—18h02 ...........................

472. 458 13. Lie–Poisson and Euler–Poincar´ Reduction e The variables ξ a may be regarded as the rigid part of the variables on the original configuration space, while xα are the internal variables. As in Simo, Lewis, and Marsden [1991], the division of variables into internal and rigid parts has deep implications for both stability theory and for bifurcation theory, again, continuing along lines developed originally by Riemann, Poincar´, and others. The main way this new insight is achieved is through e a careful split of the variables, using the (mechanical) connection as one of the main ingredients. This split puts the second variation of the augmented Hamiltonian at a relative equilibrium as well as the symplectic form into “normal form.” It is somewhat remarkable that they are simultaneously put into a simple form. This link helps considerably with an eigenvalue analysis of the linearized equations, and in Hamiltonian bifurcation theory; see, for example, Bloch, Krishnaprasad, Marsden, and Ratiu [1994a]. One of the key results in Hamiltonian reduction theory says that the reduction of a cotangent bundle T ∗ Q by a symmetry group G is a bundle over T ∗ S, where S = Q/G is shape-space, and where the fiber is either g∗ , the dual of the Lie algebra of G, or is a coadjoint orbit, depending on whether one is doing Poisson or symplectic reduction. We refer to Montgomery, Marsden, and Ratiu [1984] and Marsden [1992] and Cendra, Marsden, and Ratiu [1998]for details and references. The Lagrange–Poincar´ equations e give the analogue of this structure on the tangent bundle. Remarkably, equations (13.8.3) are very close in form to the equations for a mechanical system with classical nonholonomic velocity constraints (see Naimark and Fufaev [1972] and Koiller [1992].) The connection chosen in that case is the one-form that determines the constraints. This link is made precise in Bloch, Krishnaprasad, Marsden, and Murray [1994]. In addition, this structure appears in several control problems, especially the problem of stabilizing controls considered by Bloch, Krishnaprasad, Marsden, and Sanchez [1992]. For systems with a momentum map J constrained to a specific value µ, the key to the construction of a reduced Lagrangian system is the modification of the Lagrangian L to the Routhian Rµ , which is obtained from the Lagrangian by subtracting off the mechanical connection paired with the constraining value µ of the momentum map. On the other hand, a basic ingredient needed for the Lagrange–Poincar´ equations is a velocity shift e in the Lagrangian, the shift being determined by the connection, so this velocity-shifted Lagrangian plays the role that the Routhian does in the constrained theory. ........................... 15 July 1998—18h02 ...........................

473. This is page 459 Printer: Opaque this 14 Coadjoint Orbits In this chapter we prove, amongst other things, that the coadjoint orbits of a Lie group are symplectic manifolds. These symplectic manifolds are, in fact, the symplectic leaves for the Lie–Poisson bracket. This result was developed and used by Kirillov, Arnold, Kostant, and Souriau in the early to mid-1960s, although it had important roots going back to the work of Lie, Borel, and Weil. (See Kirillov [1962, 1976b], Arnold [1966a], Kostant [1970], and Souriau [1969].) Here we give a direct proof. Alternatively, one can give a proof using general reduction theory, as in Marsden and Weinstein [1974] and Abraham and Marsden [1978]. Recall from Chapter 9 that the adjoint representation of a Lie group G is defined by Adg = Te Ig : g → g, where Ig : G → G is the inner automorphism Ig (h) = ghg −1 . The coadjoint action is given by Ad∗−1 : g∗ → g∗ , g where Ad∗−1 is the dual of the linear map Adg−1 , that is, it is defined by g Ad∗−1 (µ), ξ = µ, Adg−1 (ξ) , g where µ ∈ g∗ , ξ ∈ g, and , denotes the pairing between g∗ and g. The coadjoint orbit, Orb(µ), through µ ∈ g∗ is the subset of g∗ defined by Orb(µ) := {Ad∗−1 (µ) | g ∈ G} := G · µ. g

474. 460 14. Coadjoint Orbits Like the orbit of any group action, Orb(µ) is an immersed submanifold of g∗ and if G is compact, Orb(µ) is a closed embedded submanifold.1 14.1 Examples of Coadjoint Orbits (a) Rotation Group. As we saw in §9.3, the adjoint action for SO(3) is AdA (v) = Av where A ∈ SO(3) and v ∈ R3 ∼ so(3). Identify so(3)∗ with R3 by the = usual dot product, that is, if Π, v ∈ R3 , we have Π, v = Π · v. Thus, for ˆ Π ∈ so(3)∗ and A ∈ SO(3), Ad∗ −1 (Π), v = Π, AdA−1 (v)ˆ = Π, (A−1 v)ˆ = Π · A−1 v A ˆ = (A−1 )T Π · v = AΠ · v (14.1.1) since A is orthogonal. Hence, with so(3)∗ identiﬁed with R3 , Ad∗ −1 = A, A and so Orb(Π) = {Ad∗ −1 (Π) | A ∈ SO(3)} = {AΠ | A ∈ SO(3)}, A (14.1.2) which is the sphere in R3 of radius Π . (b) Aﬃne Group on R. Consider the Lie group of transformations of R of the form T (x) = ax + b where a = 0. Identify G with the set of pairs (a, b) ∈ R2 with a = 0. Since (T1 ◦ T2 )(x) = a1 (a2 x + b2 ) + b1 = a1 a2 x + a1 b2 + b1 and 1 T −1 (x) = (x − b), a we take group multiplication to be (a1 , b1 ) · (a2 , b2 ) = (a1 a2 , a1 b2 + b1 ). (14.1.3) The identity element is (1, 0) and the inverse of (a, b) is 1 b (a, b)−1 = ,− . (14.1.4) a a Thus, G is a two-dimensional Lie group. It is an example of a semidirect product. (See Exercise 9.3-1.) As a set, the Lie algebra of G is g = R2 ; to 1 The coadjoint orbits are also embedded (but not necessarily closed) submanifolds of g∗ if G is an algebraic group. ........................... 15 July 1998—18h02 ...........................

475. 14.1 Examples of Coadjoint Orbits 461 compute the bracket on g we shall first compute the adjoint representation. The inner automorphisms are given by I(a,b) (c, d) = (a, b) · (c, d) · (a, b)−1 1 b = (ac, ad + b) · ,− a a = (c, ad − bc + b), (14.1.5) and so, differentiating (14.1.5) with respect to (c, d) at the identity in the direction of (u, v) ∈ g, gives Ad(a,b) (u, v) = (u, av − bu). (14.1.6) Differentiating (14.1.6) with respect to (a, b) in the direction (r, s) gives the Lie bracket [(r, s), (u, v)] = (0, rv − su). (14.1.7) An alternative approach is to realize (a, b) as the matrix a b ; 0 1 one checks that group multiplication corresponds to matrix multiplication. Then the Lie algebra, identified with matrices u v , 0 0 has the bracket given by the commutator. The adjoint orbit through (u, v) is {u}×R if (u, v) = (0, 0) and is {(0, 0)} if (u, v) = (0, 0). The adjoint orbit {u} × R cannot be symplectic, as it is one dimensional. To compute the coadjoint orbits, denote elements of g∗ by the column vector, α (α, β)T = β and use the pairing α (u, v), = αu + βv (14.1.8) β to identify g∗ with R2 . Then α α Ad∗ (a,b) , (u, v) = , Ad(a,b) (u, v) β β α = , (u, av − bu) β = αu + βav − βbu. (14.1.9) ........................... 15 July 1998—18h02 ...........................

476. 462 14. Coadjoint Orbits Thus, α α − βb Ad∗ (a,b) = . (14.1.10) β βa If β = 0, the coadjoint orbit through (α, β)T is a single point. If β = 0, the orbit through (α, β)T is R2 minus the α-axis. It is sometimes convenient to identify the dual g∗ with g, that is, with matrices α β 0 0 via paring of g∗ with g is given by the trace of matrix multiplication of an element of g∗ with the transpose conjugate of an element of g. (c) Orbits in X∗ . Let G = Diff vol (Ω), the group of volume-preserving div diffeomorphisms of a region Ω in Rn , with Lie algebra Xdiv (Ω). In Exam- ple (d) of §10.2 we identified X∗ (Ω) with Xdiv (Ω) by using the L2 -pairing div on vector fields. Here we begin by finding a different representative of the dual X∗ (Ω), which is more convenient for explicitly determining the coad- div joint action. Then we return to the identification above and will find the expression for the coadjoint action on Xdiv (Ω); it will turn out to be more complicated. The main technical ingredient used below is the Hodge decomposition theorem for manifolds with boundary. Here we state only the relevant facts to be used below. A k-form α is called tangent to ∂Ω if i∗ (∗α) = 0. Let Ωk (Ω) denote all k-forms on M which are tangent to ∂Ω. One of the Hodge t decomposition theorems states that there is an L2 -orthogonal decomposition Ωk (Ω) = dΩk−1 (Ω) ⊕ {α ∈ Ωk (Ω) | δα = 0}. t This implies that the pairing , : {α ∈ Ω1 (Ω) | δα = 0} × Xdiv (Ω) → R t given by M, X = Mi X i dn x. (14.1.11) Ω is weakly nondegenerate. Indeed, if M ∈ {α ∈ Ω1 (M ) | δα = 0} t and M, X = 0 for all X ∈ Xdiv (Ω), then M, B = 0 for all B ∈ {Ω1 (Ω) | δB = 0} t ........................... 15 July 1998—18h02 ...........................

477. 14.1 Examples of Coadjoint Orbits 463 because the index lowering operator given by the metric on Ω induces an ismorphism between Xdiv (Ω) and {α ∈ Ω1 (Ω) | δB = 0}. t Therefore, by the L2 -orthogonal decomposition quoted above, M = df and hence M = 0. Similarly, if X ∈ Xdiv (Ω) and M, X = 0 for all M ∈ {α ∈ Ω1 (M ) | δα = 0}, then M, X = 0 for all such M , and as t before X = df , that is, X = f . But this implies X = 0 since Xdiv (Ω) and gradients are L2 -orthogonal by Stokes’ theorem. Therefore, we can identify X∗ (Ω) = {M ∈ Ω1 (Ω) | δM = 0}. div t (14.1.12) The coadjoint action of Diff vol (Ω) on X∗ (Ω) is computed in the follow- div ing way. Recall from Chapter 9 that Adϕ (X) = ϕ∗ X for ϕ ∈ Diff vol (Ω) and X ∈ Xdiv (Ω). Thus, Ad∗ −1 M, X = M, Adϕ−1 X = ϕ M · ϕ∗ X dn x = ϕ∗ M · Xdn x Ω Ω by the change of variables formula. Therefore, Ad∗ −1 M = ϕ∗ M ϕ and so Orb M = {ϕ∗ M | ϕ ∈ Diff vol (Ω)}. (14.1.13) Next, let us return to the identification of Xdiv (Ω) with itself by the L2 -pairing on vector fields X, Y = X · Y dn x. (14.1.14) Ω The Helmholtz decomposition says that any vector field on Ω can be uniquely decomposed orthogonally in a sum of a gradient of a function and a divergence-free vector field tangent to ∂Ω; this decomposition is equivalent to the Hodge decomposition on one-forms quoted before. This shows that (14.1.14) is a weakly nondegenerate pairing. For ϕ ∈ Diff vol (Ω), denote by (T ϕ)† the adjoint of T ϕ : T Ω → T Ω relative to the metric (14.1.14). By the change of variables formula, Ad∗ −1 Y, X = Y, Adϕ−1 X = ϕ Y · ϕ∗ X dn x Ω = Y · (T ϕ−1 ◦ X ◦ ϕ) dn x Ω = ((T ϕ−1 )† ◦ Y ◦ ϕ) · X dn x, Ω that is, Ad∗ −1 Y = (T ϕ−1 )† ◦ Y ◦ ϕ ϕ (14.1.15) ........................... 15 July 1998—18h02 ...........................

478. 464 14. Coadjoint Orbits and Orb Y = {(T ϕ−1 )† ◦ Y ◦ ϕ | ϕ ∈ Diff vol (Ω)}. (14.1.16) This example shows that different pairings give rise to different formulas for the coadjoint action and that the choice of dual is dictated by the specific application one has in mind. For example, the pairing (14.1.14) was convenient for the Lie–Poisson bracket on Xdiv (Ω) in Example (d) of §10.2. On the other hand, many computations involving the coadjoint action are simpler with the choice (14.1.12) of the dual corresponding to the pairing (14.1.11). (d) Orbits in X∗ . Let G = Diff can (P ) be the group of canonical trans- can formations of a symplectic manifold P with H 1 (P ) = 0. Letting k be a function on P, and Xk the corresponding Hamiltonian vector field, and ϕ ∈ G, we have Adϕ Xk = ϕ∗ Xk = Xk◦ϕ−1 (14.1.17) so identifying g with F(P ) modulo constants, or equivalently with functions on P with zero average, we get Adϕ k = ϕ∗ k = k ◦ ϕ−1 . On the dual space, which is identified with F(P ) (modulo constants) via the L2 -pairing, a straightforward verification shows that Ad∗ −1 f = ϕ∗ f = f ◦ ϕ−1 . ϕ (14.1.18) One sometimes says that Orb(f ) = {f ◦ ϕ−1 | ϕ ∈ Diff can (P )} consists of canonical rearrangements of f . (e) Toda Orbit. Another interesting example is the Toda orbit, which arises in the study of completely integrable systems. Let g = Lie algebra of real n × n lower triangular matrices with trace zero, G = lower triangular matrices with determinant one, and identify g∗ = the upper triangular matrices, using the pairing ξ, µ = Trace(ξµ), where ξ ∈ g and µ ∈ g∗ . Since AdA ξ = AξA−1 , we get Ad∗ −1 µ = P (AµA−1 ), A (14.1.19) ........................... 15 July 1998—18h02 ...........................

479. 14.1 Examples of Coadjoint Orbits 465 where P : sl(n, R) → g∗ is the projection sending any matrix to its upper triangular part. Now let   0 1 0 ··· 0 0  0 0 1 ··· 0 0    0 0 0 ··· 0 0   µ= . . . . .  ∈ g∗ . . (14.1.20)  . . . .. . .  . . . . .  0 0 0 ··· 0 1  0 0 0 ··· 0 0 One finds that Orb(µ) = {P (Aµ A−1 ) | A ∈ G} consists of matrices of the form   b1 a1 0 0 ··· 0 0  0 b2 a2 0 ··· 0 0     0 0 b3 a3 ··· 0 0     0 0 0 b4 ··· 0 0  L= , (14.1.21)  . . . . .. . .   . . . . . . . . . . . . .     0 0 0 0 ··· bn−1 an−1  0 0 0 0 ··· 0 bn where bn = 0. See Kostant [1980] and Symes [1982a,b] for further information. (f ) Coadjoint Orbits That Are Not Submanifolds. We now give an example of a Lie group G, whose generic coadjoint orbits in g∗ are not submanifolds, which is due to Kirillov [1976b], p. 293. Let α be irrational, define  it    e 0 z  G =  0 eiαt w t ∈ R, z, w ∈ C , (14.1.22)   0 0 1 and note the G is diffeomorphic to R5 . As a group, it is the semidirect product of eit 0 H= t∈R 0 eiαt with C2 , the action being by left multiplication of vectors in C2 by elements of H (see Exercise 9.3-1). Thus, the identity element of G is the 3×3 identity matrix and  it −1  −it  e 0 z e 0 −ze−it  0 eiαt w =  0 e−iαt −we−iαt  . 0 0 1 0 0 1 ........................... 15 July 1998—18h02 ...........................

480. 466 14. Coadjoint Orbits The Lie algebra g of G is     it 0 x  g =  0 iαt y  t ∈ R, x, y ∈ C (14.1.23)   0 0 0 with the usual commutator bracket as Lie bracket. Identify g∗ with     is 0 0  g∗ =  0 iαs 0 s ∈ R, a, b ∈ C (14.1.24)   a b 0 via the nondegenerate pairing in gl(3, C) is given by A, B = Re (trace(AB)). The adjoint action of  it    e 0 z is 0 x g=0 eiαt w on ξ =  0 iαs y  0 0 1 0 0 0 is given by   is 0 eit x − isz Adg ξ =  0 iαs eiαt y − iαsw . (14.1.25) 0 0 0 The coadjoint action of the same group element g on   iu 0 0 µ =  0 iαu 0 a b 0 is given by   iu 0 0 Ad∗−1 µ =  0 g iαu 0 , (14.1.26) ae−it be−iαt 0 where 1 u =u+ Im(ae−it z + be−iαt αw). (14.1.27) 1 + α2 If a, b = 0, the orbit through µ is two dimensional; it is a cylindrical surface whose generator is the u -axis and whose base is the curve in C2 given parametrically by t → (ae−it , be−iαt ). This curve, however, is the irrational ﬂow on the torus with radii |a| and |b|, that is, the cylindrical surface accumulates on itself and thus is not a submanifold of R5 . In addition, note that the closure of this orbit is the three dimensional manifold which is the product of the u -line with the two-dimensional torus of radii |a| and |b|. We shall return to this example towards the the end of §14.4. ........................... 15 July 1998—18h02 ...........................

481. 14.2 Tangent Vectors to Coadjoint Orbits 467 Exercises 14.1-1. Show that for µ ∈ g∗ , −1 −1 Orb(µ) = JR JL (µ) = JL JR (µ) . 14.1-2. Work out (14.1.10) using matrix notation. 14.2 Tangent Vectors to Coadjoint Orbits In general, orbits of a Lie group action, while manifolds in their own right, are not submanifolds of the ambient manifold; they are only injectively immersed manifolds. A notable exception occurs in the case of compact Lie groups: then all their orbits are closed embedded submanifolds. Coadjoint orbits are no exception to this global problem, as we saw in the preceding examples. We shall always regard them as injectively immersed submanifolds, diffeomorphic to G/Gµ , where Gµ = {g ∈ G | Ad∗ µ = µ} is the g isotropy subgroup of the coadjoint action at a point µ in the orbit. We now describe tangent vectors to coadjoint orbits. Let ξ ∈ g and let g(t) be a curve in G tangent to ξ at t = 0; for example, let g(t) = exp(tξ). Let O be a coadjoint orbit, and µ ∈ O. If η ∈ g, then µ(t) = Ad∗ −1 µ g(t) (14.2.1) is a curve in O with µ(0) = µ. Differentiating the identity µ(t), η = µ, Adg(t)−1 η (14.2.2) with respect to t at t = 0, we get µ (0), η = − µ, adξ η = − ad∗ µ, η , ξ and so µ (0) = − ad∗ µ. ξ (14.2.3) Thus, Tµ O = {ad∗ µ | ξ ∈ g}. ξ (14.2.4) This calculation also proves that the infinitesimal generator of the coadjoint action is given by ξg∗ (µ) = − ad∗ µ. ξ (14.2.5) The following characterization of the tangent space to coadjoint orbits is often useful. We let gµ = {ξ ∈ g | ad∗ µ = 0} be the coadjoint isotropy ξ algebra of µ; it is the Lie algebra of the coadjoint isotropy group Gµ = {g ∈ G | Ad∗ µ = µ}. g ........................... 15 July 1998—18h02 ...........................

482. 468 14. Coadjoint Orbits Proposition 14.2.1. Let , : g∗ × g → R be a weakly nondegenerate pairing and let O be the coadjoint orbit through µ ∈ g∗ . Let g◦ := {ν ∈ g∗ | ν, η = 0 µ for all η ∈ gµ } be the annihilator of gµ in g∗ . Then Tµ O ⊂ g◦ . If g is finite dimensional, µ then Tµ O = g◦ . The same equality holds if g and g∗ are Banach spaces, µ Tµ O is closed in g∗ , and the pairing is strongly nondegenerate. Proof. For any ξ ∈ g, and η ∈ gµ we have ad∗ µ, η = µ, [ξ, η] = − ad∗ µ, ξ = 0, ξ η which proves the inclusion Tµ O ⊂ g◦ . If g is finite dimensional, equality µ holds since dim Tµ O = dim g − dim gµ = dim g◦ . If g and g∗ are infinite- µ dimensional Banach spaces and , : g∗ × g → R is a strong pairing, we can assume without loss of generality that it is the natural pairing between a Banach space and its dual. If g◦ = Tµ O pick ν ∈ g◦ , such that ν = 0 µ µ and ν ∈ Tµ O. By the Hahn–Banach theorem there is an η ∈ g such that ν, η = 1 and ad∗ µ, η = 0 for all ξ ∈ g. The latter condition is equivalent ξ to η ∈ gµ . On the other hand, since ν ∈ g◦ we have ν, η = 0, which is a µ contradiction. Examples of Tangent Vectors (a) Rotation Group. Identifying (so(3), [· , ·]) ∼ (R3 , ×) and so(3)∗ ∼ = = R3 via the natural pairing given by the Euclidean inner product, formula (14.2.5) reads as follows for Π ∈ so(3)∗ and ξ, η ∈ so(3): ξ so(3)∗ (Π), η = −Π · (ξ × η) = −(Π × ξ) · η (14.2.6) so that ξ so(3)∗ (Π) = −Π×ξ = ξ×Π. As expected, ξso(3)∗ (Π) ∈ TΠ Orb(Π) is tangent to the sphere Orb(Π). Allowing ξ to vary in so(3) ∼ R3 , one = obtains all of TΠ Orb(Π). (b) Affine Group on R. Let (u, v) ∈ g and consider the coadjoint orbit through the point α ∈ g∗ . β Then (14.2.5) reads α α (u, v)g∗ = , [ · , (u, v)] . (14.2.7) β β But α α , [(r, s), (u, v)] = , (0, rv − su) = rvβ − suβ, β β ........................... 15 July 1998—18h02 ...........................

483. 14.3 The Symplectic Structure on Coadjoint Orbits 469 and so α vβ (u, v)g∗ = . (14.2.8) β −uβ If β = 0, these vectors span g∗ = R2 as they should. (c) The Group Diff vol . For G = Diff vol and M ∈ X∗ , we get the tan- div gent vectors to Orb(M ) by differentiating (14.1.13) with respect to ϕ, yield- ing TM Orb(M ) = {−£v M | v is divergence free and tangent to ∂Ω}. (14.2.9) (d) The Group Diff can (P). For G = Diff can (P ), we have Tf Orb(f ) = {−{f, k} | k ∈ F(P )}. (14.2.10) (e) The Toda Lattice. The tangent space to the Toda orbit consists of matrices of the same form as L in (14.1.21) since those matrices form a linear space. The reader can check that (14.2.4) gives the same answer. Exercises 14.2-1. Show that for the affine group on R, the Lie–Poisson bracket is ∂f ∂g ∂f ∂g {f, g}(α, β) = β − . ∂α ∂β ∂β ∂α 14.3 The Symplectic Structure on Coadjoint Orbits Theorem 14.3.1 (Coadjoint Orbit Theorem). Let G be a Lie group and let O ⊂ g∗ be a coadjoint orbit. Then ω ± (µ)(ξg∗ (µ), ηg∗ (µ)) = ± µ, [ξ, η] (14.3.1) for all µ ∈ O and ξ, η ∈ g define symplectic forms on O. We refer to ω ± as the coadjoint orbit symplectic structures and, if there is danger of ± confusion, denote it ωO . ........................... 15 July 1998—18h02 ...........................

484. 470 14. Coadjoint Orbits Proof. We prove the result for ω − , the argument for ω + being similar. First we show that formula (14.3.1) gives a well-defined form; that is, the right-hand side is independent of the particular ξ ∈ g and η ∈ g which define the tangent vectors ξg∗ (µ) and ηg∗ (µ). This follows by observing that ξg∗ (µ) = ξg∗ (µ) implies − µ, [ξ, η] = − µ, [ξ , η] for all η ∈ g. Therefore, ω − (µ)(ξg∗ (µ), ηg∗ (µ)) = ω − (ξg∗ (µ), ηg∗ (µ) , so ω − is well defined. Second, we show that ω − is nondegenerate. Since the pairing , is nondegenerate, ω − (µ)(ξg∗ (µ), ηg∗ (µ)) = 0 for all ηg∗ (µ) implies − µ, [ξ, η] = 0 for all η. This means that 0 = − µ, [ξ, ·] = ξg∗ (µ). Finally, we show that ω − is closed, that is, dω − = 0. To do this we begin by defining, for each ν ∈ g∗ , the one-form νL on G by ∗ νL (g) = (Tg Lg−1 )(ν), where g ∈ G. The one-form νL is readily checked to be left invariant; that is L∗ νL = νL for all g ∈ G. For ξ ∈ g, let ξL be the corresponding left g invariant vector field on G, so νL (ξL ) is a constant function on G (whose value at any point is ν, ξ ). Choose ν ∈ O and consider the surjective map ϕν : G → O defined by g → Ad∗−1 (ν) and the two-form σ = ϕ∗ ω − on G. g ν We claim that σ = dνL . (14.3.2) To prove this, notice that (Te ϕν )(η) = ηg∗ (ν) (14.3.3) so that the surjective map ϕν is submersive at e. By definition of pull back, σ(e)(ξ, η) equals (ϕ∗ ω − )(e)(ξ, η) = ω − (ϕν (e))(Te ϕν · ξ, Te ϕν · η) ν = ω − (ν)(ξg∗ (ν), ηg∗ (ν)) = − ν, [ξ, η] . (14.3.4) Hence σ(ξL , ηL )(e) = σ(e)(ξ, η) = − ν, [ξ, η] = − νL , [ξL , ηL ] (e). (14.3.5) We shall need the relation σ(ξL , ηL ) = − νL , [ξL , ηL ] at each point of G; to get it, we first prove two lemmas. ........................... 15 July 1998—18h02 ...........................

485. 14.3 The Symplectic Structure on Coadjoint Orbits 471 Lemma 14.3.2. The map Ad∗−1 : O → O preserves ω − , that is, g (Ad∗−1 )∗ ω − = ω − . g Proof. To prove this, we recall two identities from Chapter 9. First, (Adg ξ)g∗ = Ad∗−1 ◦ξg∗ ◦ Ad∗ , g g (14.3.6) which is proved by letting ξ be tangent to a curve h(ε) at ε = 0, recalling that d Adg ξ = gh(ε)g −1 (14.3.7) dε ε=0 and noting d (Adg ξ)g∗ (µ) = Ad∗ (gh(ε)g −1 )−1 µ dε ε=0 d = Ad∗−1 Ad∗ −1 Ad∗ (µ) g h(ε) g . (14.3.8) dε ε=0 Second, we require the identity Adg [ξ, η] = [Adg ξ, Adg η], (14.3.9) which follows by diﬀerentiating the relation Ig (Ih (k)) = Ig (h)Ig (k)Ig (h−1 ) (14.3.10) with respect to h and k and evaluating at the identity. Evaluating (14.3.6) at ν = Ad∗−1 µ, we get g (Adg ξ)g∗ (ν) = Ad∗−1 ·ξg∗ (µ) = Tµ Ad∗−1 ·ξg∗ (µ), g g (14.3.11) by linearity of Ad∗−1 . Thus, g ((Ad∗−1 )∗ ω − )(µ)(ξg∗ (µ), ηg∗ (µ)) g = ω − (ν)(Tµ Ad∗−1 ·ξg∗ (µ), Tµ Ad∗−1 ·ηg∗ (µ)) g g = ω − (ν)((Adg ξ)g∗ (ν), (Adg η)g∗ (ν)) (by (14.3.11)) = − ν, [Adg ξ, Adg η] (by deﬁnition of ω − ) = − ν, Adg [ξ, η] (by (14.3.9)) = − Ad∗ ν, [ξ, η] = − µ, [ξ, η] g = ω − (µ)(ξg∗ (µ), ηg∗ (µ)). (14.3.12) ........................... 15 July 1998—18h02 ...........................

486. 472 14. Coadjoint Orbits Lemma 14.3.3. The two form σ is left invariant, that is, L∗ σ = σ for g all g ∈ G. Proof. Using the equivariance identity ϕν ◦Lg = Ad∗−1 ◦ϕν , we compute g L∗ σ = L∗ ϕ∗ ω − = (ϕν ◦ Lg )∗ ω − = (Ad∗−1 ◦ϕν )∗ ω − g g ν g = ϕ∗ (Ad∗−1 )∗ ω − = ϕ∗ ω − = σ. ν g ν Lemma 14.3.4. We have the identity σ(ξL , ηL ) = − νL , [ξL , ηL ] . Proof. Both sides are left invariant and are equal at the identity by (14.3.5). The exterior derivative dα of a one-form α is given in terms of the Jacobi– Lie bracket by (dα)(X, Y ) = X[α(Y )] − Y [α(X)] − α([X, Y ]). (14.3.13) Since νL (ξL ) is constant, ηL [νL (ξL )] = 0 and ξL [νL (ηL )] = 0, so Lemma 14.3.4 implies2 σ(ξL , ηL ) = (dνL )(ξL , ηL ). (14.3.14) Lemma 14.3.5. We have the equality σ = dνL . (14.3.15) Proof. We shall prove that for any vector ﬁelds X and Y , σ(X, Y ) = (dνL )(X, Y ). Indeed, since σ is left invariant, σ(X, Y )(g) = (L∗−1 σ)(g)(X(g), Y (g)) g = σ(e)(T Lg−1 · X(g), T Lg−1 · Y (g)) = σ(e)(ξ, η) (where ξ = T Lg−1 · X(g) and η = T Lg−1 · Y (g)) = σ(ξL , ηL )(e) = (dνL )(ξL , ηL )(e) (by (14.3.14)) ∗ = (Lg dνL )(ξL , ηL )(e) (since νL is left-invariant) = (dνL )(g)(T Lg · ξL (e), T Lg · ηL (e)) = (dνL )(g)(T Lg · ξ, T Lg · η) = (dνL )(g)(X(g), Y (g)) = (dνL )(X, Y )(g). 2 Any Lie group carries a natural connection associated to the left (or right) action. The calculation (14.3.13) is essentially the calculation of the curvature of this connection and is closely related to the Maurer–Cartan equations (see §9.1). ........................... 15 July 1998—18h02 ...........................

487. 14.3 The Symplectic Structure on Coadjoint Orbits 473 Since σ = dνL by Lemma 14.3.5, dσ = ddνL = 0, and so 0 = dϕ∗ ω − = ϕ∗ dω − . ν ν From ϕν ◦ Lg = Ad∗−1 ◦ϕν , it follows that submersivity of ϕν at e is g equivalent to submersivity of ϕν at any g ∈ G, that is, ϕν is a surjective submersion. Thus, ϕ∗ is injective, and hence dω − = 0. ν Since coadjoint orbits are symplectic, we get the following: Corollary 14.3.6. Coadjoint orbits of finite-dimensional Lie groups are even dimensional. Corollary 14.3.7. Let Gν = {g ∈ G | Ad∗−1 ν = ν} be the isotropy g subgroup of the coadjoint action of ν ∈ g∗ . Then Gν is a closed subgroup of G, and so the quotient G/Gν is a smooth manifold with smooth projection π : G → G/Gν ; g → g · Gν . We identify G/Gν ∼ Orb(ν) via the diffeomor- = phism ρ : g · Gν ∈ G/Gν → Ad∗−1 (ν) ∈ Orb(ν). Thus, G/Gν is symplectic, g with symplectic form ω − induced from dνL , that is, dνL = π ∗ ρ∗ ω − (respectively, dνR = π ∗ ρ∗ ω † ). As we shall see in Example (a) of §14.5, ω − is not exact in general, even though π ∗ ρ∗ ω − is. Examples (a) Rotation Group. Consider Orb(Π), the coadjoint orbit through Π ∈ R3 ; then ξ R3 (Π) = ξ × Π ∈ TΠ (Orb(Π)), and η R3 (Π) = η × Π ∈ TΠ (Orb(Π)), and so with the usual identification of so(3) with R3 , the (–) coadjoint orbit symplectic structure becomes ω − (ξ R3 (Π), η R3 (Π)) = −Π · (ξ × η). (14.3.16) Recall that the oriented area of the (planar) parallelogram spanned by two vectors v, w ∈ R3 , is given by v × w (the numerical area is v × w ). Thus, the oriented area spanned by ξ R3 (Π) and η R3 (Π) is (ξ × Π) × (η × Π) = [(ξ × Π) · Π] η − [(ξ × Π) · η] Π = Π(Π · (ξ × η)). The area element dA on a sphere in R3 assigns to each pair (v, w) of tangent vectors the number dA(v, w) = n · (v × w), where n is the unit ........................... 15 July 1998—18h02 ...........................

488. 474 14. Coadjoint Orbits outward normal (this is the area of the parallelogram spanned by v and w, taken “+” if v, w, n form a positively oriented basis and “−” otherwise). For a sphere of radius Π and tangent vectors v = ξ × Π and w = η × Π, we have Π dA(ξ × Π, η × Π) = · ((ξ × Π) × (η × Π)) Π Π = · ((ξ × Π) · Π)η − ((ξ × Π) · η)Π) Π = Π Π · (ξ × η). (14.3.17) Thus, 1 ω − (Π) = − dA. (14.3.18) Π The use of “dA” for the area element is, of course, a notational abuse since this two-form cannot be exact. Likewise, 1 ω † (Π) = dA. (14.3.19) Π Notice that ω † / Π = (dA)/ Π 2 is the solid angle subtended by the area element dA. (b) Affine Group on R. For α β = 0, and µ = β on the open orbit O, formula (14.3.1) gives α ω − (µ) ((r, s)g∗ (µ), (u, v)g∗ (µ)) = − , [(r, s), (u, v)] β = β(rv − su). (14.3.20) Using the coordinates, (α, β) ∈ R2 , this reads 1 ω − (µ) = dα ∧ dβ. (14.3.21) β (c) The Group Diff vol . For a coadjoint orbit of G = Diff vol (Ω) the (+) coadjoint orbit symplectic structure at a point M becomes ω † (M )(−£v M, −£w M ) = − M · [v, w] dn x, (14.3.22) Ω where [v, w] is the Jacobi–Lie bracket. Note that we have indeed a minus sign on the right-hand side of (14.3.22) since [v, w] is minus the left Lie algebra bracket. ........................... 15 July 1998—18h02 ...........................

489. 14.4 The Orbit Bracket via Restriction of the Lie–Poisson Bracket 475 Exercises 14.3-1. Let G be a Lie group. Find an action of G on T ∗ G for which the map J(ξ)(νL (g)) = − νL (g), ξL (g) = − ν, ξ is an equivariant momentum map. 14.3-2. Relate the calculations of this section to the Maurer–Cartan equations. 14.3-3. Give another proof that dω ± = 0 by showing that XH for ω ± coincides with that for the Lie–Poisson bracket and hence that Jacobi’s identity holds. 14.3-4 (The Group Diff can ). For a coadjoint orbit for G = Diff can (P ), show that the (+) coadjoint orbit symplectic structure is ω † (L)({k, f }, {h, f }) = f {k, h} dq dp. P 14.3-5 (The Toda Lattice). For the Toda orbit, check that the orbit symplectic structure is n−1 1 ω † (f ) = dbi ∧ dai . i=1 ai 14.3-6. Verify formula (14.3.21); that is, 1 ω − (µ) = dα ∧ dβ. β 14.4 The Orbit Bracket via Restriction of the Lie–Poisson Bracket Theorem 14.4.1 (Lie–Poisson-Coadjoint Orbit Compatibility). The Lie–Poisson bracket and the coadjoint orbit symplectic structure are consistent in the following sense: for F, H : g∗ → R and O a coadjoint orbit in g∗ , {F, H}+ |O = {F |O, H|O}† . (14.4.1) Here, the bracket {F, G}+ is the (+) Lie–Poisson bracket, while the bracket on the right-hand side of (14.4.1) is the Poisson bracket defined by the (+) coadjoint orbit symplectic structure on O. Similarly, {F, H}− |O = {F |O, H|O}− . (14.4.2) The following paragraph summarizes the basic content of the theorem. ........................... 15 July 1998—18h02 ...........................

490. 476 14. Coadjoint Orbits Two Approaches to the Lie–Poisson Bracket There are two different ways to produce the same Lie–Poisson bracket {F, H}− (respectively, {F, H}+ ) on g∗ : Extension Method: 1. Take F, H : g∗ → R; 2. extend F, H to FL , HL : T ∗ G → R by left (respectively, right) translation; 3. take the bracket {FL , HL } with respect to the canonical symplectic structure on T ∗ G; and 4. restrict: {FL , HL }|g∗ = {F, H}− (respectively, {FR , HR }|g∗ = {F, H}+ ). Restriction Method: 1. Take F, H : g∗ → R; 2. form the restrictions F |O, H|O to a coadjoint orbit; and 3. take the Poisson bracket {F |O, H|O}− with respect to the − (respectively, +) orbit symplectic structure ω − (respectively, ω † ) on the orbit O: for µ ∈ O we have {F |O, H|O}− (µ) = {F, H}− (µ). (respectively, {F |O, H|O}† (µ) = {F, H}+ (µ)). Proof of Theorem 14.4.1. Let µ ∈ O. By definition, δF δH {F, H}− (µ) = − µ, , . (14.4.3) δµ δµ On the other hand, {F |O, H|O}− (µ) = ω − (XF , XH )(µ), (14.4.4) where XF and XH are the Hamiltonian vector fields on O generated by F |O and H|O, and ω − is the minus orbit symplectic form. Recall that the Hamiltonian vector field XF on g∗ is given by − XF (µ) = ad∗ (µ), ξ (14.4.5) where ξ = δF /δµ ∈ g. Motivated by this, we prove the following: ........................... 15 July 1998—18h02 ...........................

491. 14.4 The Orbit Bracket via Restriction of the Lie–Poisson Bracket 477 Lemma 14.4.2. Using the orbit symplectic form ω − , for µ ∈ O we have XF |O (µ) = ad∗ δF/δµ (µ). (14.4.6) Proof. Let ξ, η ∈ g, so (14.3.1) gives ω − (µ)(ad∗ µ, ad∗ µ) = − µ, [ξ, η] = µ, adη (ξ) = ad∗ (µ), ξ . ξ η η (14.4.7) Letting ξ = δF/δµ and η be arbitrary, we get δF ω − (µ)(ad∗ ∗ δF/δµ µ, adη µ) = ad∗ µ, η = dF (µ) · ad∗ µ. η (14.4.8) δµ Thus, XF |O (µ) = ad∗ δF/δµ µ, as required. To complete the proof of Theorem 14.4.1, note that {F |O, H|O}− (µ) = ω − (µ)(XF |O (µ), XH|O (µ)) = ω − (µ)(ad∗ ∗ δF/δµ µ, adδH/δµ µ) δF δH = − µ, , = {F, H}− (µ). (14.4.9) δµ δµ Corollary 14.4.3. (i) For H ∈ F(g∗ ), the trajectory of XH starting at µ stays in Orb(µ). (ii) A function C ∈ F(g∗ ) is a Casimir function iff δC/δµ ∈ gµ for all µ ∈ g∗ . (iii) If C ∈ F(g∗ ) is Ad∗ -invariant (constant on orbits) then C is a Casimir function. The converse is also true if all coadjoint orbits are connected. Proof. Part (i) follows from the fact that XH (ν) is tangent to the coadjoint orbit O for ν ∈ O, since XH (ν) = ad∗ δH/δµ (ν). Part (ii) follows from the definitions and formula (14.4.5), and (iii) follows from (ii) by writing out the condition of Ad∗ -invariance as C(Ad∗−1 µ) = C(µ) and differentiating g in g at g = e. For the converse, recall Proposition 10.4.7 which states that any Casimir function is constant on the symplectic leaves. Thus, since the connected components of the coadjoint orbits are the symplectic leaves of g∗ , the Casimir functions are constant on them. In particular, if the coadjoint orbits are connected, the Casimir functions are constant on each coadjoint orbit which then imples that they are all Ad∗ -invariant. ........................... 15 July 1998—18h02 ...........................

492. 478 14. Coadjoint Orbits To illustrate part (iii), we note that for G = SO(3), the function 1 2 CΦ (Π) = Φ Π 2 is invariant under the coadjoint action (A, Π) → AΠ and is therefore a Casimir function. Another example is given by G = Diff can (P ), and the functional CΦ (f ) := Φ(f ) dq dp, P where dq dp is the Liouville measure and Φ is any function of one variable. This is a Casimir function since it is Ad∗ -invariant by the change of variables formula. In general, Ad∗ -invariance of C is a stronger condition than C being a Casimir function. Indeed if C is Ad∗ -invariant, differentiating the relation C(Ad∗−1 µ) = C(µ) relative to µ rather than g as we did in the proof of g (iii), we get δC δC = Adg (14.4.10) δ(Ad∗−1 µ) g δµ for all g ∈ G. Taking g ∈ Gµ , this relation becomes δC/δµ = Adg (δC/δµ), that is, δC/δµ belongs to the centralizer of Gµ in g, that is, to the set Cent(Gµ , g) := {ξ ∈ g | Adg ξ = ξ for all g ∈ Gµ }. Letting Cent(gµ , g) := {ξ ∈ g | [η, ξ] = 0 for all η ∈ gµ } denote the centralizer of gµ in g, we see, by differentiating the relation defining Cent(Gµ , g) with respect to g at the identity, that Cent(Gµ , g) ⊂ Cent(gµ , g). Thus, if C is Ad∗ -invariant, then δC ∈ gµ ∩ Cent(gµ , g) = Cent(gµ ) = the center of gµ . δµ Thus, we conclude the following: Proposition 14.4.4 (Kostant [1979]). If C is an Ad∗ -invariant function on g∗ , then δC/δµ lies in both Cent(Gµ , g) and in Cent(gµ ). If C is a Casimir function, then δC/δµ lies in the center of gµ . Proof. The first statement follows from the preceding considerations. The second statement is deduced in the following way. Let G0 be the connected component of the identity in G. Since the Lie algebras of G and of G0 coincide, a Casimir function C of g∗ is necessarily constant on the G0 - coadjoint orbits, since they are connected (see Corollary 14.4.3(iii)). Thus, by the first part, δC/δµ ∈ Cent(gµ ). ........................... 15 July 1998—18h02 ...........................

493. 14.4 The Orbit Bracket via Restriction of the Lie–Poisson Bracket 479 By the theorem of Duflo and Vergne [1969] (see Chapter 9), for generic µ ∈ g∗ , the coadjoint isotropy gµ is abelian and therefore Cent(gµ ) = gµ generically. The above corollary and proposition leave open, in principle, the possibility of non-Ad∗ -invariant Casimir functions on g∗ . This is not possible for Lie groups with connected coadjoint orbits, as we saw before. If C : g∗ → R is a function such that δC/δµ ∈ gµ for all µ ∈ g∗ , but there is at least one ν ∈ g∗ such that δC/δν ∈ Cent(gν ), then C is a Casimir / function that is not Ad∗ -invariant. This element ν ∈ g∗ must be such that its coadjoint orbit is disconnected, and it must be nongeneric. We know of no such example of a Casimir function. On the other hand, the above statements provide easily verifiable criteria for the form of, or the nonexistence of, Casimir functions on duals of Lie algebras. For example, if g∗ has open orbits whose union is dense, it cannot have Casimir functions. Indeed, any such function would have to be constant on the connected components of each orbit, and thus by continuity, on g∗ . An example of such a Lie algebra is that of the affine group on the line discussed in Example (b) of §14.1. The same argument shows that Lie algebras with at least one dense orbit have no Casimir functionals. Example The purpose of this example is to show that Casimir functions do not characterize generic coadjoint orbits. Let us use Corollary 14.4.3 to determine all Casimir functions for the Lie algebra in Example (f) of §14.1. If     iu 0 0 is 0 x µ =  0 iαu 0 ∈ g∗ , ξ =  0 iαs y  ∈ g, a b 0 0 0 0 for a, b, x, y ∈ C, u, s ∈ R, then it is straightforward to check that   iu 0 0 ad∗ µ =  0 ξ iαu 0 , −isa −iαsb 0 where 1 u =− Im(ax + αby). 1 + α2 Thus, if at least one of a, b is not zero, then     0 0 x  gµ = 0 0 y  Im(ax + αby) = 0 ,   0 0 0 whereas if a = b = 0, then gµ = g. For C : g∗ → R, denote by   iCu 0 Ca δC = 0 iαCu Cb  , δµ 0 0 0 ........................... 15 July 1998—18h02 ...........................

494. 480 14. Coadjoint Orbits where Cu ∈ R, Ca , Cb ∈ C are the partial derivatives of C relative to the variables u, a, b. Thus, the condition δC/δµ ∈ gµ for all µ implies that Cu = 0, that is, C is independent of u and Im(aCa + αbCb ) = 0. (14.4.11) The same condition could have been obtained by lengthier direct calculations involving the Lie–Poisson bracket. Here are the highlights. The commutator bracket on g is given by       is 0 x iu 0 z 0 0 i(sz − ux)  0 iαs y  ,  0 iαu w = 0 0 iα(sw − uy) , 0 0 0 0 0 0 0 0 0 so that for µ ∈ g∗ parametrized by u ∈ R, a, b, ∈ C, we have δF δH {F, H} (µ) = − Re Trace µ , δµ δµ = Im[a(Fu Ha − Hu Fa ) + αb(Fu Hb − Hu Fb )]. (14.4.12) Taking Fu = Fb = 0 in {F, C} = 0, forces Cu = 0. Then the remaining condition reduces to (14.4.11). To solve (14.4.11) we need first to convert it into a real equation. Regard C as being defined on C2 with values in C and write C = A + iB, with A and B real-valued functions. Write a = p + iq, b = v + iw so that by the Cauchy-Riemann equations we have Ap = Bq , Aq = −Bp , Av = Bw , Aw = −Bv and also, since C is holomorphic For JEM TSR: why Ca = Ap + iBp = Bq − iAq = Cp = −iCq holomorphic? Cb = Av + iBv = Bw − iAw = Cv = −iCw . Therefore, 0 = Im((p + iq)(Ap + iBp ) + α(v + iw)(Av + iBv )) = qAp + pBp + α(wAv + vBv ) = qAp − pAq + αwAv − αvAw by the Cauchy–Riemann equations. We solve this partial diffferential equation by the method of characteristics. The flow of the vector field with components (q, −p, αw, −αv) is given by Ft (p, q, v, w) = (p cos t + q sin t, −p sin t + q cos t, v cos αt + w sin αt, −v sin αt + w cos αt) ........................... 15 July 1998—18h02 ...........................

495. 14.4 The Orbit Bracket via Restriction of the Lie–Poisson Bracket 481 and thus Note: Using A = f (p2 + q 2 , v 2 + w2 ) invariants in for any real valued function f : R2 → R is the general solution of this ways we have equation. Thus, any Casimir function is a functional of p2 + q 2 and v 2 + w2 . not discussed! Note that Ca = Ap + iBp = Ap − iAq , and Cb = Av + iBv = Av − iAw . For A = p2 + q 2 , we have hence Ca = 2(p − iq) and Cb = 0. One can then A is used in 2 verify directly that p2 + q 2 is a Casimir function using formula (14.4.12). different ways! Similarly, one sees that v 2 + w2 is a Casimir function. Since the generic leaf of g∗ is two-dimensional (see Example 14.1(f)) and the dimension of g is five, it follows that the Casimir functions do not characterize the generic coadjoint orbits. This is in agreement with the obrservation made in Example 14.1(f) that the generic coadjoint orbits have as closure the three-dimensional submanifolds of g∗ , which are the product of the torus of radii |a| and |b| and the u -line, if one expresses the orbit through   iu 0 0 µ =  0 iαu 0 a b 0 as     iu 0 0   0 iu 0 u = u + Im(ae−it z + be−iαt αw), t ∈ R, z, w ∈ C .   ae−it be−iαt 0 Note that this is consistent with these two Casimir functions preserving |ae−it | = |a| and |be−iαt | = |b|. A mathematical reason coadjoint orbits and the Lie–Poisson bracket are so important is that Hamiltonian systems with symmetry are sometimes a covering of a coadjoint orbit. This is proved below. If X and Y are topological spaces, a continuous surjective map p : X → Y is called a covering map if every point in Y has an open neighborhood U such that p−1 (U ) is a disjoint union of open sets in X, called the decks over U. Note that each deck is homeomorphic to U by p. If p : M → N is a surjective proper map of smooth manifolds which is also a local diffeomorphism, then it is a covering map. For example, SU(2) (the spin group) forms a covering space of SO(3) with two decks over each point and SU(2) is simply connected while SO(3) is not. (See Chapter 9.) Transitive Hamiltonian actions have been characterized by Lie, Kostant, Kirillov, and Souriau in the following manner (see Kostant [1966]): Theorem 14.4.5 (Kostant’s Coadjoint Orbit Covering Theorem). Let P be a Poisson manifold and let Φ : G × P → P be a left, transitive, Hamiltonian action with equivariant momentum map J : P → g∗ . Then ........................... 15 July 1998—18h02 ...........................

496. 482 14. Coadjoint Orbits (i) J : P → g∗ is a canonical submersion onto a coadjoint orbit of G in + g∗ . (ii) If P is symplectic, J is a symplectic local diffeomorphism onto a coadjoint orbit endowed with the “+” orbit symplectic structure. If J is also proper, then it is a covering map. Proof. (i) That J is a canonical map was proved in §12.4. Since Φ is transitive, choosing a z0 ∈ P, any z ∈ P can be written as z = Φg (z0 ) for some g ∈ G. Thus, by equivariance J(P ) = {J(z) | z ∈ P } = {J(Φg (z0 )) | g ∈ G} = {Ad∗−1 J(z0 ) | g ∈ G} = Orb(J(z0 )). g Again by equivariance, for z ∈ P we have Tz J(ξP (z)) = − ad∗ J(z), which ξ has the form of a general tangent vector at J(z) to the orbit Orb(J(z0 )); thus, J is a submersion. (ii) If P is symplectic with symplectic form Ω, J is a symplectic map if the orbit has the “+” symplectic form: ω † (µ)(ad∗ µ, ad∗ µ) = µ, [ξ, η] . This is ξ η seen in the following way. Since Tz P = {ξP (z) | ξ ∈ g} by transitivity of the action, (J∗ ω † )(z)(ξP (z), ηP (z)) = ω † (J(z))(Tz J(ξP (z)), Tz J(ηP (z))) = ω † (J(z))(ad∗ J(z), ad∗ J(z)) ξ η = J(z), [ξ, η] = J([ξ, η])(z) = {J(ξ), J(η)}(z) (by equivariance) = Ω(z)(XJ(ξ) (z), XJ(η) (z)) = Ω(z)(ξP (z), ηP (z)), (14.4.13) which shows that J∗ ω † = Ω, that is, J is symplectic. Since any symplectic map is an immersion, J is a local diffeomorphism. If J is also proper, it is a symplectic covering map, as discussed above. If J is proper and the symplectic manifold P is simply connected, the covering map in (ii) is a diffeomorphism; this follows from classical theorems about covering spaces (Spanier [1966]). It is clear that if Φ is not transitive, J(P ) is a union of coadjoint orbits. See Guillemin and Sternberg [1984] and Grigore and Popp [1989] for more information. Exercises 14.4-1. Show that if C is a Casimir function on a Poisson manifold, then {F, K}C = C{F, K} is also a Poisson structure. If XH is a Hamiltonian vector field for { , }, show that it is also Hamiltonian for { , }C with the Hamiltonian function CH. ........................... 15 July 1998—18h02 ...........................

497. 14.5 The Special Linear Group on the Plane 483 14.4-2. Does Kostant’s coadjoint orbit covering theorem ever apply to group actions on cotangent bundles by cotangent lift? 14.5 The Special Linear Group on the Plane In the Lie algebra sl(2, R) of traceless real 2 × 2 matrices, introduce the basis 0 1 0 0 1 0 e= , f= , h= . 0 0 1 0 0 −1 Note that [h, e] = 2e, [h, f ] = −2f , and [e, f ] = h. Identify sl(2, R) with R3 via ξ := xe + yf + zh ∈ sl(2, R) → (x, y, z) ∈ R3 . (14.5.1) The nonzero structure constants are c3 = 1, c1 = −2, and c2 = 2. We 12 13 23 identify the dual space sl(2, R)∗ with sl(2, R) via the nondegenerate pairing trace(αξ T ?) α, ξ = trace(αξ). (14.5.2) In particular, the dual basis of {e, f , h} is {f , e, 1 h} and we identify sl(2, R)∗ 2 with R3 using this basis, that is, 1 α = af + be + c h → (a, b, c) ∈ R3 . (14.5.3) 2 The (±) Lie–Poisson bracket on sl(2, R)∗ is thus given by δF δH {F, H}± (α) = ± trace α , , δα δα where δF δF δα, = trace δα = DF (α) · δα δα δα d = F (α + tδα) dt t=0 ∂F ∂F ∂F = δa + δb + δc, ∂a ∂b ∂c and where 1 ∂F ∂F 2 δc δb δF ∂c ∂a δα = and = . δa − 1 δc 2 δα ∂F − ∂F ∂b ∂c ........................... 15 July 1998—18h02 ...........................

498. 484 14. Coadjoint Orbits The expression of the Lie–Poisson bracket in coordinates is therefore ∂F ∂G ∂F ∂G ∂F ∂G ∂F ∂G {F, G}± (a, b, c) = 2a − ± 2b − ∂a ∂c ∂c ∂a ∂b ∂c ∂c ∂b ∂F ∂G ∂F ∂G ±c − . (14.5.4) ∂a ∂b ∂b ∂a Since SL(2, R) is connected, the Casimir functions are the Ad∗ -invariant functions on sl(2, R)∗ . Since Adg ξ = gξg −1 , if g ∈ SL(2, R) and ξ ∈ sl(2, R), it follows that Ad∗−1 α = gαg −1 , g for α ∈ sl(2, R)∗ . The determinant of 1 2c b a −1c 2 is obviously invariant under conjugation. Therefore, for R3 endowed with Tudor: logic the (±) Lie–Poisson bracket of sl(2, R)∗ , any function of the form was back- wards? 1 C(a, b, c) = Φ ab + c2 (14.5.5) 4 for a C 1 function Φ : R → R is a Casimir function. The symplectic leaves are in fact the sheets of the hyperboloids 1 1 C0 (a, b, c) := ab + c2 = constant = 0, (14.5.6) 2 4 the two nappes (without vertex) of the cone ab + (1/4)c2 = 0, and the origin. One can verify this directly by using Ad∗−1 ξ = gξg −1 . The orbit g symplectic structure on these hyperboloids is given by ω − (a, b, c)(ad∗ ∗ (x,y,z) (a, b, c), ad(x ,y ,z ) (a, b, c)) = −a(2zx − 2xz ) − b(2yz − 2zy ) − c(xy − yx ) 1 =− (area element of the hyperboloid). (14.5.7) C0 (a, b, c) To prove the last equality in (14.5.7), use the formulas ad∗ (x,y,z) (a, b, c) = (2az − cy, cx − 2bz, 2by − 2zx), ad∗ ∗ (x,y,z) (a, b, c) × ad(x ,y ,z ) (a, b, c) = 2bc(xy − yx ) + 4b2 (yz − zy ) + 4ab(zx − xz ), 2ac(xy − yx ) + 4ab(yz − zy ) + 4a2 (zx − xz ), c2 (xy − yx ) + 2bc(yz − zy ) + 2ac(zx − xz ) , ........................... 15 July 1998—18h02 ...........................

499. 14.6 The Euclidean Group of the Plane 485 and the fact that (ab + 1 c2 ) = (b, a, 1 c) is normal to the hyperboloid to 4 2 get, as in (14.3.18), dA(a, b, c)(ad∗ ∗ (x,y,z) (a, b, c), ad(x ,y ,z ) (a, b, c)) (b, a, 1 c) = 2 · (ad∗ ∗ (x,y,z) (a, b, c) × ad(x ,y ,z ) (a, b, c)) (b, a, 1 c) 2 =− C0 (a, b, c) )ω − (a, b, c)(ad∗ ∗ (x,y,z) (a, b, c), ad(x ,y ,z ) (a, b, c)). Exercises 14.5-1. Using traces, ﬁnd a Casimir function for sl(3, R)∗ . 14.6 The Euclidean Group of the Plane We use the notation and terminology from Exercise 11.5-2. Recall that the Check exercise group SE(2) consists of matrices of the form x-ref. Rθ a (Rθ , a) := , (14.6.1) 0 1 where a ∈ R2 and Rθ is the rotation matrix cos θ − sin θ Rθ = , (14.6.2) sin θ cos θ The identity element is the 3 × 3 identity matrix and the inverse is given by −1 Rθ a R−θ −R−θ a = . (14.6.3) 0 1 0 1 The Lie algebra se(2) of SE(2) consists of 3 × 3 block matrices of the form −ωJ v , (14.6.4) 0 0 where 0 1 J= , (14.6.5) −1 0 (note, as usual, that JT = J−1 = −J,) with the usual commutator bracket. If we identify se(2) with R3 by the isomorphism −ωJ v ∈ se(2) → (ω, v) ∈ R3 (14.6.6) 0 0 ........................... 15 July 1998—18h02 ...........................

500. 486 14. Coadjoint Orbits the expression for the Lie algebra bracket becomes [(ω, v1 , v2 ), (ζ, w1 , w2 )] = (0, ζv2 − ωw2 , ωw1 − ζv1 ) = (0, ωJT w − ζJT v), (14.6.7) where v = (v1 , v2 ) and w = (w1 , w2 ). The adjoint action of Rθ a −ωJ v (Rθ , a) = on (ω, v) = 0 1 0 0 is given by conjugation Rθ a −ωJ v Rθ −Ra −ωJ −ωJa + Rθ v = (14.6.8) 0 1 0 0 0 1 0 0 or, in coordinates, Ad(Rθ ,v) (ω, v) = (ω, −ωJa + Rθ v) . (14.6.9) In proving this, we used the identity Rθ J = JRθ . Identify se(2)∗ with matrices of the form 2J µ 0 (14.6.10) α 0 via the nondegenerate pairing given by the trace of the produce. Thus, se(2)∗ is isomorphic to R3 via 2J µ 0 ∈ se(2)∗ → (µ, α) ∈ R3 , (14.6.11) α 0 so that in these coordinates the pairing between se(2)∗ and se(2) becomes (µ, α), (ω, v) = µω + α · v, (14.6.12) that is, the usual dot product in R3 . The coadjoint action is thus given by Ad(Rθ ,a)−1 (µ, α) = (µ + Rθ α · Ja, Rθ α) . (14.6.13) Indeed, by (14.6.3), (14.6.5), (14.6.9), (14.6.12), and (14.6.13) we get Ad∗ θ ,a)−1 (µ, α), (ω, v) = (µ, α), Ad(Rθ ,−Rθ a) (ω, v) (R = (µ, α), (ω, ωJR−θ a + R−θ v) = µω + ωα · JR−θ a + α · R−θ v = (µ + α · R−θ Ja) ω + Rθ α · v = (µ + Rθ α · Ja, Rθ α) , (ω, v) . ........................... 15 July 1998—18h02 ...........................

501. 14.6 The Euclidean Group of the Plane 487 Coadjoint Orbits in se(2)∗ . Formula (14.6.13) shows that the coadjoint orbits are the cylinders T ∗ Sα = {(µ, α) | α = constant} , 1 if α = 0 and the points are on the µ-axis. The canonical cotangent bundle projection is denoted by π : T ∗ Sα −→ Sα , 1 1 π(µ, α) = α. Connectedness of SE(2) implies by Corollary 14.4.3(iii) that the Casimir functions coincide with the functions invariant under the coadjoint action (14.6.13), that is, all Casimir functions have the form 1 2 C(µ, α) = Φ 2 α (14.6.14) for a smooth function Φ : [0, ∞) → R. The Lie–Poisson Bracket on se(2)∗ . We next determine the (±) Lie– Poisson bracket on se(2)∗ . If F : se(2)∗ ∼ R × R2 → R, its functional = derivative is δF ∂F = , αF , (14.6.15) δ(µ, α) ∂µ where (µ, α) ∈ se(2)∗ ∼ R × R2 and α F denotes the gradient of F with = respect to α. The (±) Lie–Poisson structure on se(2)∗ is given by ∂F ∂G {F, G}± (µ, α) = ± Jα · αG − Jα · αF . (14.6.16) ∂µ ∂µ It can now be directly veriﬁed that the functions given by (14.6.14) are indeed Casimir functions for the bracket (14.6.16) The Symplectic Form on Orbits. The coadjoint action of se(2) on se(2)∗ is given by ad(ξ, u)∗ (µ, α) = (−Jα · u, ξJα) (14.6.17) On the coadjoint orbit representing a cylinder about the µ-axis, the orbit symplectic structure is ω(µ, α)(ad(ξ, u)∗ (µ, α), ad(η, v)∗ (µ, α)) = ±(ξJα · v − ηJα · u) = ±(area element dA on the cylinder)/ α . (14.6.18) The last equality is proved in the following way. Since the outward unit normal to the cylinder is (0, α)/ α , by (14.6.17) the area element dA is ........................... 15 July 1998—18h02 ...........................

502. 488 14. Coadjoint Orbits given by dA(µ, α)((−Jα · u, ξJα), (−Jα · v, ηJα)) (0, α) = · [((−Jα · u, ξJα) × (−Jα · u, ξJα)] α = α (ξJα · v − ηJα · u). Let us show that on the orbit through (µ, α), the symplectic form α ω − , the canonical symplectic form of the cotangent bundle T ∗ Sα . Since π(µ, α) = 1 α, it follows by (14.6.17) that T(µ,α) π (ad(ξ, α)∗ (µ, α)) = ξJα, thought of as a tangent vector to S 1 at α. The length of this vector is |ξ| α , so we identify it with the pair (ξ α , α) ∈ Tα Sα . The canonical 1 one-form is given by Θ(µ, α) · ad(ξ, α)∗ (µ, α) = (µ, α) · T(µ,α) π (ad(ξ, α)∗ (µ, α) = (µ, α) · (ξ α , α) = µξ α . (14.6.19) To compute the canonical symplectic form Ω on T ∗ S 1 in these notations, we extend the tangent vectors ad(ξ, u)∗ (µ, α) ad(η, v)∗ (µ, α) ∈ T(µ,α) Sα ∗ 1 to vector ﬁelds X : (µ, α) → ad(ξ, u)∗ (µ, α) and Y : (µ, α) → ad(η, v)∗ (µ, α) and use (14.6.17) to get ad(ξ, α)∗ (µ, α) · [θ(Y )](µ, α) = dθ(Y )(µ, α) · ad(ξ, α)∗ (µ, α) d = θ(Y )(µ(t), α(t)), dt t=0 where (µ(t), α(t)) is a curve in T ∗ Sα such that 1 (µ(0), α(0)) = (µ, α) and (µ (0), α (0)) = ad(η, v)∗ (µ, α). Since α(t) = α , we conclude that this is equal to d µ(t)η α = µ (0)η α = −Jα · uη α . (14.6.20) dt t=0 ........................... 15 July 1998—18h02 ...........................

503. 14.6 The Euclidean Group of the Plane 489 Similarly, ad(η, v)∗ (µ, α) · (θ(X))(µ, α) = −Jα · vξ α . (14.6.21) Finally, since X = (ξ, u)se(2)∗ , Y = (η, v)se(2)∗ , we conclude that [X, Y ](µ, α) = −[(ξ, u), (η, v)]se(2)∗ (µ, α) = −(0, ξJT v − ηJT u)se(2)∗ (µ, α) ∗ = − ad 0, ξJT v − ηJT u (µ, α) and by ( 14.6.19) that θ([X, Y ])(µ, α) = 0. (14.6.22) Therefore, by (14.6.21), and (14.6.22) we get Ω(µ, α) (ad(ξ, u)∗ (µ, α), ad(η, v)∗ (µ, α), ) = −dθ(X, Y )(µ, α) = − ad(ξ, u)∗ (µ, α) · [θ(Y )](µ, α) + ad(η, v)∗ (µ, α) · [θ(X)](µ, α) + θ([X, y])(µ, α) = − α ηJα · u + α ξJα · v = α (ξJα · v − ηJα · u) which shows that Ω = α ω − = area form on the cylinder of radius α . Lie Algebra Deformations. The Poisson structures of so(3)∗ , sl(2, R)∗ , and se(2)∗ ﬁt together in a larger Poisson manifold. Weinstein [1983b] considers for every ε ∈ R the Lie algebra gε with abstract basis X1 , X2 , X3 and relations [X3 , X1 ] = X2 , [X2 , X3 ] = X1 , [X1 , X2 ] = εX3 . (14.6.23) If ε 0, the map √ √ X1 → ε(1, 0, 0)∧ , X2 → ε(0, 1, 0)∧ , X3 → (0, 0, 1)∧ , (14.6.24) deﬁnes an isomorphism of gε with so(3), while if ε = 0, the map X1 → (0, 0, −1), X2 → (0, −1, 0), X3 → (−1, 0, 0), (14.6.25) ........................... 15 July 1998—18h02 ...........................

504. 490 14. Coadjoint Orbits defines an isomorphism of g0 with se(2), and if ε 0, the map √ √ −ε 1 0 −ε 0 1 1 0 −1 X1 → , X2 → , X3 → , 2 0 −1 2 1 0 2 1 0 (14.6.26) defines an isomorphism of gε with sl(2, R). The (+) Lie–Poisson structure of g∗ is given by the bracket relations ε {x3 , x1 } = x2 , {x2 , x3 } = x1 , {x1 , x2 } = εx3 , (14.6.27) for the coordinate functions xi ∈ g∗ = R3 , xi , xj = δij . ε In R4 with coordinate functions (x1 , x2 , x3 , ε) consider the above bracket relations plus {ε, x1 } = {ε, x2 } = {ε, x3 } = 0. This defines a Poisson structure on R4 which is not of Lie–Poisson type. The leaves of this Poisson structure are all two dimensional in the space (x1 , x2 , x3 ) and the Casimir functions are all functions of x2 + x2 + εx2 and 1 2 3 ε. The inclusion of g∗ in R4 with the above Poisson structure is a canonical ε map. The leaves of R4 with the above Poisson structure as ε passes through zero is given in Figure 14.6.1. 14.7 The Euclidean Group of Three-Space The Euclidean Group, its Lie Algebra and its Dual. An element of SE(3) is a pair (A, a) where A ∈ SO(3) and a ∈ R3 ; the action of SE(3) on R3 is the rotation A followed by translation by the vector a and has the expression (A, a) · x = Ax + a. (14.7.1) Using this formula, one sees that multiplication and inversion in SE(3) are given by (A, a)(B, b) = (AB, Ab + a) (14.7.2) and (A, a)−1 = (A−1 , −A−1 a), (14.7.3) for A, B ∈ SO(3) and a, b ∈ R3 . The identity element is (l, 0). Note that SE(3) embeds into SL(4; R) via the map A a (A, a) → ; (14.7.4) 0 1 ........................... 15 July 1998—18h02 ...........................

505. 14.7 The Euclidean Group of Three-Space 491 µ Π3 ε0 Π2 α2 ε=0 Π1 α1 a–b ε0 c a+b Figure 14.6.1. The orbit structure for so(3)∗ , se(2)∗ , and sl(2, R)∗ . thus one can operate with SE(3) as one would with matrix Lie groups by using this embedding. In particular, the Lie algebra se(3) of SE(3) is isomorphic to a Lie subalgebra of sl(4; R) with elements of the form ˆ x y , where x, y ∈ R3 , (14.7.5) 0 0 and a Lie algebra bracket equal to the commutator bracket of matrices. This shows that the Lie bracket operation on se(3) is given by [(x, y), (x , y )] = (x × x , x × y − x × y). (14.7.6) Since −1 A a A−1 −A−1 a = 0 1 0 1 ........................... 15 July 1998—18h02 ...........................

506. 492 14. Coadjoint Orbits and A a ˆ x y A−1 −A−1 a 0 1 0 0 0 1 Aˆ A−1 x −Aˆ A−1 a + Ay x = , 0 0 we see that the adjoint action of SE(3) on se(3) has the expression Ad(A,a) (x, y) = (Ax, Ay − Ax × a). (14.7.7) The (6 × 6)-matrix of Ad(A,a) is given by A 0 . (14.7.8) ˆ aA A Identifying the dual of se(3) with R3 ×R3 by the dot product in every factor, it follows that the matrix of Ad∗ (A,a)−1 is given by the inverse transpose of the (6 × 6)-matrix (14.7.8), that is, it equals A ˆ aA . (14.7.9) 0 A Thus, the coadjoint action of SE(3) on se(3)∗ = R3 × R3 has the expression Ad∗ (A,a)−1 (u, v) = (Au + a × Av, Av). (14.7.10) (This Lie algebra is a semidirect product and all formulas derived here “by hand” are special cases of general ones that may be found in works on semidirect products; see, for example, Marsden, Ratiu, and Weinstein [1984a,b].) Coadjoint Orbits in se(3)∗ . Let {e1 , e2 , e3 , f1 , f2 , f3 } be an orthonormal basis of se(3) = R3 × R3 such that ei = fi , i = 1, 2, 3. The dual basis of se(3)∗ via the dot product is again {e1 , e2 , e3 , f1 , f2 , f3 }. Let e and f denote arbitrary vectors satisfying e ∈ span{e1 , e2 , e3 } and f ∈ span{f1 , f2 , f3 }. For the coadjoint action the only zero-dimensional orbit is the origin. Since se(3) is six dimensional, there can also be two- and four-dimensional coadjoint orbits. These in fact occur and fall into three types. Type I: The orbit through (e, 0) equals SE(3) · (e, 0) = {(Ae, 0) | A ∈ SO(3)} = S 2e , (14.7.11) the two-sphere of radius e . Type II: The orbit through (0, f ) is given by SE(3) · (0, f ) = {(a × Af , Af )|A ∈ SO(3), a ∈ R3 } = {(u, Af )|A ∈ SO(3), u ⊥ Af } = T S 2f , (14.7.12) ........................... 15 July 1998—18h02 ...........................

507. 14.7 The Euclidean Group of Three-Space 493 the tangent bundle of the two-sphere of radius f ; note the vector part is in the first slot. Type III: The orbit through (e, f ), where e, f = 0, equals SE(3) · (e, f ) = {(Ae + a × Af , Af ) | A ∈ SO(3), a ∈ R3 }. (14.7.13) We will prove below that this orbit is diffeomorphic to T S 2f . Consider the smooth map e·f ϕ : (A, a) ∈ SE(3) → Ae + a × Af − Af , Af ∈ T S 2f (14.7.14) f 2 which is right invariant under the isotropy group SE(3)(e,f ) = {(B, b) | Be + b × f = e, Bf = f } (14.7.15) (see (14.7.10)), that is, ϕ((A, a)(B, b)) = ϕ(A, a) for all (A, a) ∈ SE(3) and (B, b) ∈ SE(3)(e,f ) . Thus, ϕ induces a smooth map ϕ : SE(3)/ SE(3)(e,f ) → T S 2f . The map ϕ is injective, for if ϕ(A, a) = ¯ ¯ ϕ(A , a ), then (A, a)−1 (A , a ) = (A−1 A , A−1 (a − a)) ∈ SE(3)(e,f ) as is easily checked. To see that ϕ (and hence ϕ ) is surjective, let (u, v) ∈ ¯ T S 2f , that is, v = f and u · v = 0. Then choose an A ∈ SO(3) such that Af = v and let a = [v × (u − Ae)]/ f 2 . It is then straightforward to check that ϕ(A, a) = (u, v) by (14.7.14). Thus, ϕ is a bijective map. Since ¯ the derivative of ϕ at (A, a) in the direction T(I,0) L(A,a) (ˆ , y) = (Aˆ , Ay) x x equals d x T(A,a) ϕ(Aˆ , Ay) = x ϕ(Aetˆ , a + tAy) dt t=0 = (A(x × e + y × f ) + a × A(x × f ) e·f − A(x × f ), A(x × f )) (14.7.16) f 2 we see that its kernel consists of left translates by (A, a) of {(x, y) ∈ se(3) | x × e + y × f = 0, x × f = 0}. (14.7.17) However, taking the derivatives of the defining relations in (14.7.15) at (B, b) = (l, 0) we see that (14.7.17) coincides with se(3)(e,f ) . This shows that ϕ is an immersion and hence, since dim(SE(3)/ SE(3)(e,f ) ) = dim T S 2f ........................... 15 July 1998—18h02 ...........................

508. 494 14. Coadjoint Orbits = 4, it follows that ϕ is a local diffeomorphism. Therefore, ϕ is a diffeomorphism. To compute the tangent spaces to these orbits, we use Proposition 14.2.1 which states that the annihilator of the coadjoint isotropy subalgebra at µ equals Tµ O. The coadjoint action of the Lie algebra se(3) on its dual se(3)∗ is computed to be ad∗ (u, v) = (u × x + v × y, v × x). (x,y) (14.7.18) Thus, the isotropy subalgebra se(3)(u,v) is given again by (14.7.17), that is, it equals {(x, y) ∈ se(3) | u × x + v × y = 0, v × x = 0}. Let O denote a nonzero coadjoint orbit in se(3)∗ . Then the tangent space at a point in O is given as follows for each of the three types of orbits: Type I: Since se(3)(e,0) = {(x, y) ∈ se(3) | e × x = 0} = span(e) × R3 , (14.7.19) it follows that the tangent space to O at (e, 0) is the tangent space to the sphere of radius e at the point e in the first factor. Type II: Since se(3)(0,f ) = {(x, y) ∈ se(3) | f × y = 0, f × x = 0} = span(f ) × span(f ), (14.7.20) it follows that the tangent space to O at (0, f ) equals f ⊥ × f ⊥ , where f ⊥ denotes the plane perpendicular to f . Type III: Since se(3)(e,f ) = {(x, y) ∈ se(3) | e × x + f × y = 0 and f × x = 0} = {(c1 f , c1 e + c2 f ) | c1 , c2 ∈ R}, (14.7.21) the tangent space at (e, f ) to O is the orthogonal complement of the space spanned by (f , e) and (0, f ), that is, it equals {(u, v) | u · f + v · e = 0 and v · f = 0}. The Symplectic Form on Orbits. Let O denote a nonzero orbit of se(3)∗ . We consider the different oribt types separately, as above. Type I: If O contains a point of the form (e, 0), the orbit O equals S 2e × {0}. The minus orbit symplectic form is ω − (e, 0)(ad∗ (e, 0), ad∗ (e, 0)) = −e · (x × x ). (x,y) (a,b) (14.7.22) ........................... 15 July 1998—18h02 ...........................

509. 14.7 The Euclidean Group of Three-Space 495 Thus, the symplectic form on O at (e, 0) is −1/ e times the area element of the sphere of radius e (see (14.3.16) and (14.3.18)). Type II: If O contains a point of the form (0, f ), then O equals T S 2f . Let (u, v) ∈ O, that is, v = f and u ⊥ v. The symplectic form in this case is ω − (u, v)(ad∗ (u, v), ad∗ (u, v)) (x,y) (a,b) = −u · (x × x ) − v · (x × y − x × y). (14.7.23) We shall prove below that this form is exact, namely, ω − = −dθ, where θ(u, v) · ad∗ (u, v) = u · x. (x,y) (14.7.24) First, note that θ is indeed well defined, for if ad∗ (u, v) = ad∗ ,y ) (u, v), (x,y) (x by (14.7.18) we have (x −x ) ×v = 0, that is, x −x = cv for some constant c ∈ R, and since u⊥v, we conclude from here that u · x = u · x . Second, in order to compute dθ, we shall use the formula dθ(X, Y ) = X[θ(Y )] − Y [θ(X)] − θ([X, Y ]) for any vector fields X, Y on O. Third, we shall choose X and Y as follows: X(u, v) = (x, y)se(3)∗ (u, v) = − ad∗ (u, v), (x,y) Y (u, v) = (x , y )se(3)∗ (u, v) = − ad∗ ,y ) (u, v), (x for fixed x, y, x , y ∈ R3 . Fourth, to compute X[θ(Y )](u, v), consider the path (u( ), v( )) = (e− x u + (v × y), e− x v), ˆ ˆ which satisfies (u(0), v(0)) = (u, v) and (u (0), v (0)) = (u × x + v × y, v × x) = ad∗ (u, v). (x,y) Then d X[θ(Y )](u, v) = θ(Y )(u( ), v( )) d =0 d = u( ) · x = (u × x + v × y) · x . d =0 ........................... 15 July 1998—18h02 ...........................

510. 496 14. Coadjoint Orbits Similarly, Y [θ(X)](u, v) = (u × x + v × y ) · x. Finally, [X, Y ](u, v) = [(x, y)se(3)∗ , (x , y )se(3)∗ ](u, v) = −[(x, y), (x , y )]se(3)∗ (u, v) = −(x × x , x × y − x × y)se(3)∗ (u, v) = ad∗(x×x ,x×y −x ×y) (u, v). Therefore, − dθ(u, v)(ad∗ (u, v), ad∗ ,y ) (u, v)) (x,y) (x = −X[θ(Y )](u, v) + Y [θ(X)](u, v) + θ([X, Y ])(u, v) = −(u × x + v × y) · x + (u × x + v × y ) · x + u · (x × x ) = −u · (x × x ) − v · (x × y − x × y), which coincides with (14.7.23). The form θ given by (14.7.24) is the canonical symplectic structure when we identify T S 2f with T ∗ S 2f using the Euclidean metric. Type III: If O contains (e, f ), where e = 0 and f = 0, then O is diffeomorphic to T ∗ S 2f in the following way. The map ϕ : SE(3) → T ∗ S 2f given by (14.7.14) induces a diffeomorphism ϕ : SE(3)/ SE(3)(e,f ) → T ∗ S 2f . However, the orbit O through (e, f ) is diffeomorphic to SE(3)/ SE(3)(e,f ) by the diffeomorphism (A, a) → Ad∗ (A,a)−1 (e, f ). (14.7.25) Therefore, the diffeomorphism Φ : O → T ∗ S 2f is given by Φ(Ad∗ (A,a)−1 (e, f )) = Φ(Ae + a × Af , Af ) (14.7.26) e·f = (Ae + a × Af − Af , Af ). f 2 If (u, v) ∈ O, the orbit symplectic structure is given by formula (14.7.23), where u = Ae + a × Af , v = Af for some A ∈ SO(3), a ∈ R3 . Let e·f e·f u = Ae + a × Af − Af = u − v, f 2 f 2 v = Af = v, (14.7.27) the pair of vectors (u, v) representing an element of T S 2 . Note that v = f and u · v = 0. Then a tangent vector to T S 2f at (u, v) can be represented as ad∗ (u, v) = (u × x + v × y, v × x) so that by (14.7.26) we (x,y) ........................... 15 July 1998—18h02 ...........................

511. 14.7 The Euclidean Group of Three-Space 497 get d T(u,v) Φ−1 (ad∗ (u, v)) = (x,y) Φ−1 (e− x u + (v × y), e x v) ˆ ˆ d =0 d e·f − x = e− x u + (v × y) + ˆ e ˆ v, e− x v ˆ d =0 f 2 e·f = u×x+v×y+ (v × x), v × x f 2 = (u × x + v × y, v × x) = ad∗ (u, v). (x,y) Therefore, the push-forward of the orbit symplectic form ω − to T S 2f is (Φ∗ ω − )(u, v)(ad∗ (u, v), ad∗ ,y ) (u, v)) (x,y) (x = ω − (u, v)(T(u,v) Φ−1 (ad∗ (u, v)), T(u,v) Φ−1 (ad∗ ,y ) (u, v)) (x,y) (x = ω − (u, v)(ad∗ (u, v), ad∗ ,y ) (u, v)) (x,y) (x = −u · (x × x ) − v · (x × y − x × y) e·f = −u · (x × x ) − v · (x × y − x × y) − v · (x × x ). f 2 (14.7.28) The first two terms represent the canonical symplectic structure on T S 2f (identified via the Euclidean metric with T ∗ S 2f ), as we have seen in the analysis of type II orbits. The third term is the following two-form on T S 2f e·f β(u, v) ad∗ (u, v), ad∗ ,y ) (u, v) = − (x,y) (x v · (x × x ). (14.7.29) f 2 As in the case of θ for type II orbits, it is easily seen that (14.7.28) correctly defines a two-form on T S 2f . It is necessarily closed since it is the difference between Φ∗ ω − and the canonical two-form on T S 2f . The two-form β is a magnetic term in the sense of §6.6. We remark that the semidirect product theory of Marsden, Ratiu, and Weinstein [1984a,b], combined with cotangent bundle reduction theory, (see, for example, Marsden [1992]) can be used to give an alternative approach to the computation of the orbit symplectic forms. We refer to Mars- den, Misiolek, Perlmutter, and Ratiu [1998] for details. Exercises 14.7-1. Let K be a quadratic form on R3 and let K be the associated symmetric (3 × 3)-matrix. Let {F, L}K = − K · ( F × L). ........................... 15 July 1998—18h02 ...........................

512. 498 14. Coadjoint Orbits Show that this is the Lie–Poisson bracket for the Lie algebra structure [u, v]K = K(u × v). What is the underlying Lie group? 14.7-2. Determine the coadjoint orbits for the Lie algebra in the preceding exercise and calculate the orbit symplectic structure. Specialize to the case SO(2, 1). 14.7-3. Classify the coadjoint orbits of SU(1, 1), namely, the group of complex (2 × 2) matrices of determinant one, of the form a b g= b a where |a|2 − |b|2 = 1. 14.7-4. The Heisenberg group is deﬁned as follows. Start with the commutative group R2 with its standard symplectic form ω, the usual area form on the plane. Form the group H = R2 ⊕ R with multiplication (u, α)(v, β) = (u + v, α + β + ω(u, v)). Note that the identity element is (0, 0) and the inverse of (u, α) is given by (u, α)−1 = (−u, −α). Compute the coadjoint orbits of this group. ........................... 15 July 1998—18h02 ...........................

513. This is page 499 Printer: Opaque this 15 The Free Rigid Body As an application of the theory developed so far, we discuss the motion of a free rigid body about a fixed point. We begin with a discussion of the kinematics of rigid body motion. Our description of the kinematics of rigid bodies follows some of the notations and conventions of continuum mechanics, as in Marsden and Hughes [1983]. 15.1 Material, Spatial, and Body Coordinates Consider a rigid body, free to move in R3 . A reference configuration B of the body is the closure of an open set in R3 with a piecewise smooth boundary. Points in B, denoted X = (X 1 , X 2 , X 3 ) ∈ B relative to an orthonormal basis (E1 , E2 , E3 ) are called material points and X i , i = 1, 2, 3, are called material coordinates. A configuration of B is a mapping ϕ : B → R3 which is, for our purposes, C 1 , orientation preserving, and invertible on its image. Points in the image of ϕ are called spatial points and denoted by lowercase letters. Let (e1 , e2 , e3 ) be a right-handed orthonormal basis of R3 . Coordinates for spatial points, such as x = (x1 , x2 , x3 ) ∈ R3 , i = 1, 2, 3, relative to the basis (e1 , e2 , e3 ) are called spatial coordinates. See Fig- ure 15.1-1. Dually, one can consider material quantities such as maps defined on B, say Z : B → R. Then we can form spatial quantities by composition: zt = Zt ◦ ϕ−1 . Spatial quantities are also called Eulerian t quantities and material quantities are often called Lagrangian quanti-

514. 500 15. The Free Rigid Body ties. A motion of B is a time-dependent family of configurations, written e3 ϕ E3 X x e2 E2 e1 E1 Figure 15.1.1. Configurations, spatial and material points. x = ϕ(X, t) = ϕt (X) or simply x(X, t) or xt (X). Spatial quantities are functions of x, and are typically written as lowercase letters. By composition with ϕt , spatial quantities become functions of the material points X. Rigidity of the body means that the distances between points of the body are fixed as the body moves. We shall assume that no external forces act on the body and that the center of mass is fixed at the origin (see Exercise 15.1-1). Since any isometry of R3 that leaves the origin fixed is a rotation (a 1932 theorem of Mazur and Ulam), we can write x(X, t) = R(t)X, i.e., xi = Ri (t)X j , j i, j = 1, 2, 3, sum on j, where xi are the components of x relative to the basis e1 , e2 , e3 fixed in space, and [Ri ] is the matrix of R relative to the basis (E1 , E2 , E3 ) and j (e1 , e2 , e3 ). The motion is assumed to be continuous and R(0) is the identity, so det(R(t)) = 1 and thus R(t) ∈ SO(3), the proper orthogonal group. Thus, the configuration space for the rotational motion of a rigid body may be identified with SO(3). Consequently, the velocity phase space of the free rigid body is T SO(3) and the momentum phase space is the cotangent bundle T ∗ SO(3). Euler angles, discussed shortly, are the traditional way to parametrize SO(3). In addition to the material and spatial coordinates, there is a third set, the convected or body coordinates. These are the coordinates associated with the moving basis, and the description of the rigid body motion in these coordinates, due to Euler, becomes very simple. As before, let E1 , E2 , E3 be an orthonormal basis fixed in the reference configuration. Let the time- dependent basis ξ 1 , ξ 2 , ξ 3 be defined by ξ i = R(t)Ei , i = 1, 2, 3, so ξ 1 , ξ 2 , ξ 3 move attached to the body. The body coordinates of a vector in R3 are its components relative to ξ i . For the rigid body anchored at the origin and rotating in space, (e1 , e2 , e3 ) is thought of as a basis fixed in space, whereas ........................... 15 July 1998—18h02 ...........................

515. 15.2 The Lagrangian of the Free Rigid Body 501 (ξ 1 , ξ 2 , ξ 3 ) is a basis fixed in the body and moving with it. For this reason (e1 , e2 , e3 ) is called the spatial coordinate system and (ξ1 , ξ 2 , ξ 3 ) the body coordinate system. See Figure 15.1-2. ϕ e3 ξ3 E3 X ξ1 x e2 E2 e1 ξ2 E1 A space fixed frame A body fixed frame Figure 15.1.2. Spatial and body frames. Exercises 15.1-1. Start with SE(3) as the configuration space for the rigid body and “reduce out” (see §10.7, the Euler–Poincar´, and Lie–Poisson reduction e theorems) translations to arrive at SO(3) as the configuration space. 15.2 The Lagrangian of the Free Rigid Body If X ∈ B is a material point of the body, the corresponding trajectory followed by X in space is x(t) = R(t)X, where R(t) ∈ SO(3). The material or Lagrangian velocity V (X, t) is defined by ∂x(X, t) ˙ V (X, t) = = R(t)X, (15.2.1) ∂t while the spatial or Eulerian velocity v(x, t) is defined by v(x, t) = V (X, t) = R(t)R(t)−1 x. ˙ (15.2.2) Finally, the body or convective velocity V(X, t) is defined by taking the velocity regarding X as time-dependent and x fixed, that is, we write ........................... 15 July 1998—18h02 ...........................

516. 502 15. The Free Rigid Body X(x, t) = R(t)−1 x, and define ∂X(x, t) V(X, t) = − = R(t)−1 R(t)R(t)−1 x ˙ ∂t = R(t)−1 R(t)X ˙ = R(t)−1 V (X, t) = R(t)−1 v(x, t). (15.2.3) See Figure 15.2.1. Assume that the mass distribution of the body is de- V(X, t) = R(t)–1v(x, t) R(t) e3 E3 X x e2 E2 e1 E1 V(X, t) = v(x, t) Figure 15.2.1. Material velocity V , spatial velocity v, and body velocity V. scribed by a compactly supported density measure ρ0 d3 X in the reference configuration, which is zero at points outside the body. The Lagrangian, taken to be the kinetic energy, is given by any of the following expressions that are related to one another by a change of variables and the identities V = V = v : 1 2 L= ρ0 (X) V (X, t) d3 X (material) (15.2.4) 2 B 1 = ρ0 (R(t)−1 x) v(x, t) 2 d3 x (spatial) (15.2.5) 2 R(t)B 1 = ρ0 (X) V(X, t) 2 d3 X (convective or body). (15.2.6) 2 B Differentiating R(t)T R(t) = Identity and R(t)R(t)T = Identity with respect to t, it follows that both R(t)−1 R(t) and R(t)R(t)−1 are skew- ˙ ˙ symmetric. Moreover, by (15.2.2), (15.2.3), and the classical definition v = ω × r = ωr ˆ of angular velocity, it follows that the vectors ω(t) and Ω(t) in R3 defined by ω(t) = R(t)R(t)−1 ˆ ˙ (15.2.7) ........................... 15 July 1998—18h02 ...........................

517. 15.3 The Lagrangian and Hamiltonian in Body Representation 503 and Ω(t) = R(t)−1 R(t) ˆ ˙ (15.2.8) represent the spatial and convective angular velocities of the body. Note that ω(t) = R(t)Ω(t), or as matrices, ω = AdR Ω = RΩR−1 . ˆ ˆ ˆ Let us show that L : T SO(3) → R given by (15.2.4) is left-invariant. Indeed, if B ∈ SO(3), left translation by B is ˙ ˙ LB R = BR and T LB (R, R) = (BR, BR) , so ˙ 1 ˙ 2 L(T LB (R, R)) = ρ0 (X) BRX d3 X 2 B 1 ˙ 2 ˙ = ρ0 (X) RX d3 X = L(R, R) (15.2.9) 2 B since R is orthogonal. By Lie–Poisson reduction of dynamics (Chapter 13), the corresponding Hamiltonian system on T ∗ SO(3), which is necessarily also left invariant, induces a Lie–Poisson system on so(3)∗ and this system leaves invariant the coadjoint orbits Π = constant. Alternatively, by Euler–Poincar´ re- e duction of dynamics, we get a system of equations in terms of body angular velocity on so(3). ˆ Reconstruction of the dynamics on T SO(3) is simply this: given Ω(t), determine R(t) ∈ SO(3) from (15.2.8): ˙ ˆ R(t) = R(t)Ω(t), (15.2.10) which is a time-dependent linear equation for R(t). 15.3 The Lagrangian and Hamiltonian for the Rigid Body in Body Representation From (15.2.6), (15.2.3), and (15.2.8) of the previous section, the rigid body Lagrangian is 1 L= ρ0 (X) Ω × X 2 d3 X. (15.3.1) 2 B ........................... 15 July 1998—18h02 ...........................

518. 504 15. The Free Rigid Body Introducing the new inner product a, b := ρ0 (X)(a × X) · (b × X) d3 X, B which is determined by the density ρ0 (X) of the body, (15.3.1) becomes 1 L(Ω) = Ω, Ω . (15.3.2) 2 In what follows, it is useful to keep in mind the identity (a × X) · (b × X) = (a · b) X 2 − (a · X)(b · X). Define the linear isomorphism l : R3 → R3 by la · b = a, b for all a, b ∈ R3 ; this is possible and uniquely determines l, since both the dot product and , are nondegenerate bilinear forms (assuming the rigid body is not concentrated on a line). It is clear that l is symmetric with respect to the dot product and is positive-definite. Let (E1 , E2 , E3 ) be an orthonormal basis for material coordinates. The matrix of l is   − ρ (X)X i X j d3 X, i = j,   0 B lij = Ei · lEj = Ei , Ej =    ρ0 (X)( X 2 − (X i )2 ) d3 X, i = j, B which are the classical expressions of the matrix of the inertia tensor . If c is a unit vector, c, c is the (classical) moment of inertia about the axis c. Since l is symmetric, it can be diagonalized; an orthonormal basis in which it is diagonal is a principal axis body frame and the diagonal elements I1 , I2 , I3 are the principal moments of inertia of the rigid body. In what follows we work in a principal axis reference and body frame, (E1 , E2 , E3 ). Since so(3)∗ and R3 are identified by the dot product (not by , ), the linear functional Ω, · —the Legendre transformation of Ω—on so(3) ∼ = R3 is identified with lΩ := Π ∈ so(3)∗ ∼ R3 because Π · a = Ω, a for all = a ∈ R3 . With l = diag(I1 , I2 , I3 ), (15.3.2) defines a function 1 Π2 1 Π2 Π2 K(Π) = + 2+ 3 (15.3.3) 2 I1 I2 I3 that represents the expression for the kinetic energy on so(3)∗ ; note that Π = lΩ is the angular momentum in the body frame. Indeed, for any a ∈ R3 , the identity (X × (Ω × X)) · a = (Ω × X) · (a × X) and the classical expression of the angular momentum in the body frame, namely, (X × V)ρ0 (X) d3 X (15.3.4) B ........................... 15 July 1998—18h02 ...........................

519. 15.3 The Lagrangian and Hamiltonian in Body Representation 505 gives (X × V)ρ0 (X) d3 X ·a= (X × (Ω × X)) · aρ0 (X) d3 X B B = (Ω × X) · (a × X)ρ0 (X) d3 X B = Ω, a = lΩ · a = Π · a, that is, the expression (15.3.4) equals Π. The angular momentum in space has the expression π= (x × v)ρ(x) d3 x, (15.3.5) R(B) where ρ(x) = ρ0 (X) is the spatial mass density and v = ω × x is the spatial velocity (see (15.2.2) and (15.2.7)). For any a ∈ R3 , π·a= (x × (ω × x)) · aρ(x) d3 X R(B) = (ω × x) · (a × x)ρ(x) d3 X. (15.3.6) R(B) Changing variables x = RX, (15.3.6) becomes (ω × RX) · (a × RX)ρ0 (X) d3 X B = (RT ω × X) · (RT a × X)ρ0 (X) d3 X B = Ω, RT a = Π · RT a = RΠ · a, that is, π = RΠ. (15.3.7) Since L given by (15.3.2) is left invariant on T SO(3), the function K deﬁned on so(3)∗ by (15.3.3) deﬁnes the Lie–Poisson equations of motion on so(3)∗ relative to the bracket {F, H}(Π) = −Π · ( F (Π) × H(Π)). (15.3.8) Since K(Π) = l−1 Π, we get from (15.3.8) the rigid body equations Π = − K(Π) × Π = Π × l−1 Π, ˙ (15.3.9) ........................... 15 July 1998—18h02 ...........................

520. 506 15. The Free Rigid Body that is, they are the standard Euler equations: ˙ I2 − I3 Π1 = Π2 Π3 , I2 I3 ˙ I3 − I1 Π2 = Π1 Π3 , (15.3.10) I1 I3 ˙ I1 − I2 Π3 = Π1 Π2 . I1 I2 The fact that these equations preserve coadjoint orbits amounts, in this case, to the easily verified fact that Π2 := Π 2 (15.3.11) is a constant of the motion. In terms of coadjoint orbits, these equations are Hamiltonian on each sphere in R3 with Hamiltonian function K. The functions 1 2 CΦ (Π) = Φ Π , (15.3.12) 2 for any Φ : R → R, are easily seen to be Casimir functions. The conserved momentum resulting from left invariance is the spatial angular momentum: π = RΠ. (15.3.13) Using left invariance, or a direct calculation, one finds that π is constant in time. Indeed, ˙ ˙ ˙ π = (RΠ)˙ = RΠ + RΠ = ω × RΠ + RΠ ˙ ˙ −1 ˙ = RΩ × RΠ + RΠ = R(−Π × l Π + Π) = 0. The flow lines are given by intersecting the ellipsoids K = constant with the coadjoint orbits which are two-spheres. For distinct moments of inertia I1 I2 I3 , or I1 I2 I3 the flow on the sphere has saddle points at (0, ±Π, 0) and centers at (±Π, 0, 0), (0, 0, ±Π). The saddles are connected by four heteroclinic orbits, as indicated in Figure 15.3.1. In §15.10 we prove: Theorem 15.3.1 (Rigid Body Stability Theorem). In the motion of a free rigid body, rotation around the long and short axes are (Liapunov ) stable and rotation about the middle axis is unstable. Even though we completely solved the rigid body equations in body representation, the actual configuration of the body, that is, its attitude in space, has not been determined yet. This will be done in §15.8. Also, one has to be careful about the meaning of stability in space versus material versus body representation. ........................... 15 July 1998—18h02 ...........................

521. 15.4 Kinematics on Lie Groups 507 Π3 Π2 Π1 Figure 15.3.1. Rigid body flow on the angular momentum spheres for the case I1 I 2 I 3 . Euler’s equations are very general. The n-dimensional case has been treated by Mishchenko and Fomenko [1976, 1978a], Adler and van Mo- erbeke [1980a,b], and Ratiu [1980, 1981, 1982] in connection with Lie algebras and algebraic geometry. The Russian school has generalized these equations further to a large class of Lie algebras and proved their complete integrability in a long series of papers starting in 1978; see the treatise of Fomenko and Trofimov [1989] and references therein. 15.4 Kinematics on Lie Groups We now generalize the notation used for the rigid body to any Lie group. This abstraction unifies ideas common to rigid bodies, fluids, and plasmas in a consistent way. If G is a Lie group, and H : T ∗ G → R is a Hamiltonian, ∗ we say it is described in the material picture. If α ∈ Tg G, its spatial representation is defined by ∗ αS = Te Rg (α), (15.4.1) while its body representation is ∗ αB = Te Lg (α). (15.4.2) ........................... 15 July 1998—18h02 ...........................

522. 508 15. The Free Rigid Body Similar notation is used for T G; if V ∈ Tg G, we get V S = Tg Rg−1 (V ) (15.4.3) and V B = Tg Lg−1 (V ). (15.4.4) Thus, we get body and space isomorphisms as follows: LeftTranslate RightTranslate (Body) G × g∗ ←−−−− T ∗ G −−−−− −−−− − −−−−−→ G × g∗ (Space). Thus, αS = Ad∗−1 αB g (15.4.5) and V S = Adg V B . (15.4.6) Part of the general theory of Chapter 13 says that if H is left (respectively, right) invariant on T ∗ G, it induces a Lie–Poisson system on g∗ (respec- − tively, g∗ ). + Exercises 15.4-1 (Cayley–Klein parameters.). Recall that the Lie algebras of SO(3) and SU(2) are the same. Recall also that SU(2) acts symplectically on C2 by multiplication of (complex) matrices. Use this to produce a momentum map J : C2 → su(2)∗ ∼ R3 . = (a) Write down J explicitly. (b) Verify by hand that J is a Poisson map. (c) If H is the rigid body Hamiltonian, compute HCK = H ◦ J. (d) Write down Hamilton’s equations for HCK and discuss the collective Hamiltonian theorem in this context. (e) Find this material, and relate it to the present context in one of the standard books (Whittaker, Pars, Hamel, or Goldstein, for example). 15.5 Poinsot’s Theorem Recall from §15.3 that the spatial angular momentum vector π is constant under the ﬂow of the free rigid body. Also, if ω is the angular velocity in space, then ω · π = Ω · Π = 2K (15.5.1) ........................... 15 July 1998—18h02 ...........................

523. 15.5 Poinsot’s Theorem 509 is a constant. From this, it follows that ω moves in an (affine) plane perpendicular to the fixed vector π, called the invariable plane. The distance from the origin to this plane is 2K/ π , hence the equation of this plane is u · π = 2K. See Figure 15.5.1. The ellipsoid of inertia in the body π ω invariable plane 2K/ π Figure 15.5.1. The invariable plane is orthogonal to π is defined by E = {Ω ∈ R3 | Ω · lΩ = 2K}. The ellipsoid of inertia in space is R(E) = {u ∈ R3 | u · RlR−1 u = 2K}, where R = R(t) ∈ SO(3) denotes the configuration of the body at time t. Theorem 15.5.1 (Poinsot’s Theorem). The moment of inertia ellipsoid in space rolls without slipping on the invariable plane. Proof. First, note that ω ∈ R(E) if ω has energy K. Next, we determine the planes perpendicular to the fixed vector π and tangent to R(E). See Figure 15.5.2. To do this, note that R(E) is the level set of 1 ϕ(u) = u · RIR−1 u 2 so that at ω ϕ(ω) = RIR−1 ω = RIΩ = RΠ = π. ........................... 15 July 1998—18h02 ...........................

524. 510 15. The Free Rigid Body Thus, the tangent plane to R(E) at ω is the invariable plane. Since the point of tangency is ω, which is the instantaneous axis of rotation, its velocity is zero, that is, the rolling of the inertia ellipsoid on the invariable plane takes place without slipping. π invariable plane ω inertia ellipsoid Figure 15.5.2. The geometry of Poinsot’s theorem. Exercises 15.5-1. Prove a generalization of Poinsot’s theorem to any Lie algebra g as follows. Assume that l : g → R is a quadratic Lagrangian; that is, a map of the form 1 l(ξ) = ξ, Aξ 2 where A : g → g∗ is a (symmetric) isomorphism. Deﬁne the energy ellipsoid with value E0 to be E0 = {ξ ∈ g | l(ξ) = E0 }. If ξ(t) is a solution of the Euler–Poincar´ equations and e g(t)−1 g(t) = ξ(t), ˙ ........................... 15 July 1998—18h02 ...........................

525. 15.6 Euler Angles 511 with g(0) = e, call the set Et = g(t)(E0 ) the energy ellipsoid at time t. Let µ = Aξ be the body momentum and µS to be changed in µS = Ad∗−1 µ g solution manual the conserved spatial momentum. Define the invariable plane to be the affine plane I = ξ(0) + {ξ ∈ g | µS , ξ = 0}, where ξ(0) is the initial condition. (a) Show that ξ S (t) = Adg(t) ξ(t), the spatial velocity, lies in I for all t; that is, I is invariant. (b) Show that ξ S (t) ∈ Et and that the surface Et is tangent to I at this point. (c) Show in a precise sense that Et rolls without slipping on the invariable plane. 15.6 Euler Angles In what follows, we adopt the conventions of Arnold [1989], Cabannes [1962], Goldstein [1980], and Hamel [1949]; these are different from the ones used by the British school (Whittaker [1927] and Pars [1965]). Let (x1 , x2 , x3 ) and (χ1 , χ2 , χ3 ) denote the components of a vector written in the basis (e1 , e2 , e3 ) and (ξ 1 , ξ 2 , ξ 3 ), respectively. We pass from the basis (e1 , e2 , e3 ) to the basis (ξ 1 , ξ 2 , ξ 3 ) by means of three consecutive counterclockwise rotations (see Figure 15.6.1). First rotate (e1 , e2 , e3 ) by an angle ϕ around e3 and denote the resulting basis and coordinates by (e1 , e2 , e3 ) and (x1 , x2 , x3 ), respectively. The new coordinates (x 1 , x 2 , x 3 ) are expressed in terms of the old coordinates (x1 , x2 , x3 ) of the same point by  1    1  x cos ϕ sin ϕ 0 x  x 2  =  − sin ϕ cos ϕ 0   x2  . (15.6.1) x3 0 0 1 x3 Denote the change of basis matrix (15.6.1) in R3 by R1 . Second, rotate (e1 , e2 , e3 ) by the angle θ around e1 and denote the resulting basis and coordinate system by (e1 , e2 , e3 ) and (x 1 , x 2 , x 3 ), respectively. The new ........................... 15 July 1998—18h02 ...........................

526. 512 15. The Free Rigid Body z 3 θ 2 y O ϕ ψ x 1 N Figure 15.6.1. Euler angles. coordinates (x 1 , x 2 , x 3 ) are expressed in terms of the old coordinates (x 1 , x 2 , x 3 ) by     1  x 1 1 0 0 x  x 2  =  0 cos θ sin θ   x 2  . (15.6.2) x 3 0 − sin θ cos θ x3 Denote the change of basis matrix in (15.6.2) by R2 . The e1 -axis, that is, the intersection of the (e1 , e2 )-plane with the (e1 , e2 )-plane is called the line of nodes and is denoted by ON . Finally, rotate by the angle ψ around e3 . The resulting basis is (ξ 1 , ξ 2 , ξ 3 ) and the new coordinates (χ1 , χ2 , χ3 ) are expressed in terms of the old coordinates (x 1 , x 2 , x 3 ) by  1     χ cos ψ sin ψ 0 x 1  χ2  =  − sin ψ cos ψ 0   x 2  . (15.6.3) χ3 0 0 1 x 3 Let R3 denote the change of basis matrix in (15.6.3). The rotation R sending (x1 , x2 , x3 ) to (χ1 , χ2 , χ3 ) is described by the matrix P = R3 R2 R1 given by   cosψ cosϕ − cosθ sinϕ sinψ cosψ sinϕ + cosθ cosϕ sinψ sinθ sinψ  − sinψ cosϕ − cosθ sinϕ cosψ − sinψ sinϕ + cosθ cosϕ cosψ sinθ cosψ  . sinθ sinϕ − sinθ cosϕ cosθ ........................... 15 July 1998—18h02 ...........................

527. 15.7 The Hamiltonian of the Free Rigid Body in the Material Description via Euler Angles 513 Thus, χ = Px; equivalently, since the same point is expressed in two ways 3 3 as i=1 χi ξ i = j=1 xi ej , we get   3 3 3 3 3 3 xj ej = χi ξ i =  Pij xj  ξ i = xj Pij ξ i , j=1 i=1 i=1 j=1 j=1 i=1 that is, 3 ej = Pij ξ i , (15.6.4) i=1 and hence P is the change of basis matrix between the rotated basis (ξ 1 , ξ 2 , ξ 3 ), and the fixed spatial basis (e1 , e2 , e3 ). On the other hand, (15.6.4) represents the matrix expression of the rotation RT sending ξ j to ej , that is, the matrix [R]ξ of R in the basis (ξ 1 , ξ 2 , ξ 3 ) is P T : 3 [R]ξ = PT , i.e., Rξ i = Pij ξ j . (15.6.5) i=1 Consequently, the matrix [R]e of R in the basis (e1 , e2 , e3 ) is given by P : 3 [R]e = P, i.e., Rej = Pij ei . (15.6.6) i=1 It is straightforward to check that if 0 ≤ ϕ 2π, 0 ≤ ψ 2π, 0 ≤ θ π, there is a bijective map between the (ϕ, ψ, θ) variables and SO(3). However, this bijective map does not define a chart, since its differential vanishes, for example, at ϕ = ψ = θ = 0. The differential is nonzero for 0 ϕ 2π, 0 ψ 2π, 0 θ π, and on this domain, the Euler angles do form a chart. 15.7 The Hamiltonian of the Free Rigid Body in the Material Description via Euler Angles The Hamiltonian of the Free Rigid Body To express the kinetic energy in terms of Euler angles, we choose the basis E1 , E2 , E3 of R3 in the reference configuration to equal the basis (e1 , e2 , e3 ) ........................... 15 July 1998—18h02 ...........................

528. 514 15. The Free Rigid Body of R3 in the spatial coordinate system. Thus, the matrix representation of R(t) in the basis ξ 1 , ξ 2 , ξ 3 equals PT , where P is given by (15.6). In this way, ω and Ω have the following expressions in the basis ξ 1 , ξ 2 , ξ 3 :  ˙ ˙   ˙  θ cos ϕ + ψ sin ϕ sin θ θ cos ψ + ϕ sin ψ sin θ ˙ ω =  θ sin ϕ − ψ cos ϕ sin θ  , Ω =  −θ sin ψ + ϕ cos ψ sin θ  . ˙ ˙ ˙ ˙ ϕ+ψ ˙ ˙ cos θ ˙ ϕ cos θ + ψ ˙ (15.7.1) By definition of Π, it follows that  ˙  I1 (ϕ sin θ sin ψ + θ cos ψ) ˙ Π =  I2 (ϕ sin θ cos ψ − θ sin ψ)  . ˙ ˙ (15.7.2) ˙ I3 (ϕ cos θ + ψ) ˙ This expresses Π in terms of coordinates on T (SO(3)). Since T (SO(3)) and T ∗ (SO(3)) are to be identified by the metric defined as the left invariant metric given at the identity by , , the variables (pϕ , pψ , pθ ) canonically conjugate to (ϕ, ψ.θ) are given by the Legendre transformation pϕ = ∂K/∂ ϕ, ˙ ˙ pψ = ∂K/∂ ψ, ˙ pθ = ∂K/∂ θ, where the expression of the kinetic energy on T (SO(3)) is obtained by plugging (15.7.2) into (15.3.3). We get ˙ pϕ = I1 (ϕ sin θ sin ψ + θ cos ψ) sin θ sin ψ ˙ ˙ ˙ + I2 (ϕ sin θ cos ϕ − θ sin ψ) sin θ cos ψ + I3 (ϕ cos θ + ψ) cos θ, ˙ ˙ ˙ pψ = I3 (ϕ cos θ + ψ), ˙ ˙ pθ = I1 (ϕ sin θ sin ψ + θ cos ψ) cos ψ ˙ ˙ ˙ − I2 (ϕ sin θ cos ψ − θ sin ψ) sin ψ, (15.7.3) whence, by (15.7.2),   ((pϕ − pψ cos θ) sin ψ + pθ sin θ cos ψ)/ sin θ Π =  ((pϕ − pψ cosθ) cos ψ − pθ sin θ sin ψ)/ sin θ  , (15.7.4) pψ and so by (15.3.3) we get the coordinate expression of the kinetic energy in the material picture to be K(ϕ, ψ, θ, pϕ , pψ , pθ ) 1 [(pϕ − pψ cos θ) sin ψ + pθ sin θ cos ψ]2 = 2 I1 sin2 θ [(pϕ − pψ cos θ) cos ψ − pθ sin θ sin ψ]2 p2 ψ + + . (15.7.5) I1 sin2 θ I3 ........................... 15 July 1998—18h02 ...........................

529. 15.7 The Hamiltonian of the Free Rigid Body in the Material Description via Euler Angles 515 This expression for the kinetic energy has an invariant expression on the cotangent bundle T ∗ (SO(3)). In fact, 1 1 K(αR ) = Ω, Ω = Tr(lR−1 RR−1 R), ˙ ˙ (15.7.6) 2 4 ∗ where αR ∈ TR (SO(3)) is deﬁned by α, Rˆ = Ω, v for all v ∈ R3 . v The equation of motion (15.3.9) can also be derived “by hand” without appeal to Lie–Poisson or Euler–Poincar´ reduction as follows. Hamilton’s e canonical equations ∂K ˙ ∂K ˙ ∂K , ϕ= ˙ , ψ= , θ= ∂pϕ ∂pψ ∂pθ ∂K ∂K ∂K pϕ = − ˙ , pψ = − ˙ , pθ = − ˙ , ∂ϕ ∂ψ ∂θ in a chart given by the Euler angles, become after direct substitution and a somewhat lengthy calculation, ˙ Π = Π × Ω. For F, G : T ∗ (SO(3)) → R, that is, F, G are functions of (ϕ, ψ, θ, pϕ , pψ , pθ ) in a chart given by Euler angles, the standard canonical Poisson bracket is ∂F ∂G ∂F ∂G ∂F ∂G {F, G} = − + ∂ϕ ∂pϕ ∂pϕ ∂ϕ ∂ψ ∂pψ ∂F ∂G ∂F ∂G ∂F ∂G − + − . (15.7.7) ∂pψ ∂ψ ∂θ ∂pθ ∂pθ ∂θ A computation shows that after the substitution (ϕ, ψ, θ, pϕ , pψ , pθ ) → (Π1 , Π2 , Π3 ), this becomes {F, G}(Π) = −Π · ( F (Π) × G(Π)) (15.7.8) which is the (−) Lie–Poisson bracket. This provides a direct check on the Lie–Poisson reduction theorem in Chapter 13. Thus (15.7.4) deﬁnes a canonical map between Poisson manifolds. The apparently “miraculous” groupings and cancellations of terms that occur in this calculation should make the reader appreciate the general theory. Exercises 15.7-1. Verify that (15.7.8), namely, {F, G}(Π) = −Π · ( F (Π) × G(Π)) holds by a direct calculation using substitution and the chain rule. ........................... 15 July 1998—18h02 ...........................

530. 516 15. The Free Rigid Body 15.8 The Analytical Solution of the Free Rigid Body Problem We now give the analytical solution of the Euler equations. These formulas are useful when, for example, one is dealing with perturbations leading to chaos via the Poincar´-Melnikov method, as in Ziglin [1980a,b], Holmes e and Marsden [1983], and Koiller [1985]. For the last part of this section, the reader is assumed to be familiar with Jacobi’s elementary elliptic functions; see, for example, Lawden [1989]. Let us make the following simplifying notations I2 − I3 I3 − I1 I1 − I2 a1 = ≥ 0, a2 = ≤ 0, and a3 = ≥ 0, I2 I3 I1 I3 I1 I2 where we assume I1 ≥ I2 ≥ I3 0. Then Euler’s equations Π = Π × l−1 Π ˙ can be written as ˙ Π1 = a1 Π2 Π3 , ˙ Π2 = a2 Π3 Π1 , (15.8.1) ˙ Π3 = a3 Π1 Π2 . For the analysis that follows it is important to recall that the angular momentum in space is fixed and that the instantaneous axis of rotation of the body in body coordinates is given by the angular velocity vector Ω. Case 1. I1 = I2 = I3 . Then a1 = a2 = a3 = 0 and we conclude that Π, and thus Ω are both constant. Hence the body rotates with constant angular velocity about a fixed axis. In Figure 15.3.1, all points on the sphere become fixed points. Case 2. I1 = I2 I3 . Then a3 = 0 and a2 = −a1 . Since a3 = 0 it follows from (15.8.1) that Π3 = constant, and thus denoting λ = −a1 Π3 we get a2 Π3 = λ. Thus, (15.8.1) become ˙ Π1 + λΠ2 = 0, ˙ Π2 − λΠ1 = 0, which has solution for initial data given at time t = 0 given by Π1 = Π1 (0) cos λt − Π2 (0) sin λt, Π2 = Π2 (0) cos λt + Π1 (0) sin λt. These formulas say that the axis of symmetry OZ of the body rotates relative to the body with angular velocity λ. It is straightforward to check that OZ, Ω, and Π are in the same plane and that Π and Ω make constant ........................... 15 July 1998—18h02 ...........................

531. 15.8 The Analytical Solution of the Free Rigid Body Problem 517 angles with OZ and thus among themselves. In addition, since I1 = I2 , we have 2 Π21 Π2 2 Π2 3 Ω = 2 + I2 + I2 I1 2 3 Π2 Π2 Π2 1 Π2 1 1 = 1 + 2 + 3 − 3 − I1 I2 I3 I1 I3 I1 I3 2 2K a2 Π3 = − = constant. I1 I3 Therefore, the corresponding spatial objects Oz (the symmetry axis of the inertia ellipsoid in space), ω, and π enjoy the same properties and hence the axis of rotation in the body (given by Ω) makes a constant angle with the angular momentum vector that is fixed in space, and thus the axis of rotation describes a right circular cone of constant angle in space. At the same time, the axis of rotation in the body (given by Ω) makes a constant angle with Oz, thus tracing a second cone in the body. See Figure 15.8.1. Consequently, the motion can be described by the rolling of a cone of constant angle in the body on a second cone of constant angle fixed in space. Whether the cone in the body rolls outside or inside the cone in space is determined by the sign of λ. Since Oz, ω, and π remain coplanar during the motion, ω and Oz rotate about the fixed vector π with the same angular velocity, namely, the component of ω along π in the decomposition of ω relative to π and the Oz-axis. This angular velocity is called the angular velocity of precession. Let e denote the unit vector along Oz and write ω = απ + βe. Therefore, 2K = ω · π = α π 2 + βe · π = α π 2 + βΠ3 , Π3 = Ω3 = ω · e = απ · e + β = αΠ3 + β, I3 and β = −a2 Π3 , so that α = 1/I1 and β = −a2 Π3 . Therefore, the angular velocity of precession equals ΠS /I1 . Figure 15.8.1. The geometry for integrating Euler’s equations. On the Π-sphere, the dynamics reduce to two fixed points surrounded by oppositely oriented periodic lines of latitude and separated by an equator of fixed points. A similar analysis applies if I1 I2 = I3 . ........................... 15 July 1998—18h02 ...........................

532. 518 15. The Free Rigid Body Case 3. I1 I2 I3 . The two integrals of energy and angular momentum Π2 1 Π2 Π3 + 2 + 3 = 2h = ab2 , (15.8.2) I1 I2 I3 Π2 + Π2 + Π2 = Π 1 2 3 2 = a2 b2 , (15.8.3) where a = Π 2 /2h, b = 2h/ Π are positive constants, enable us to express Π1 and Π3 in terms of Π2 as I1 (I2 − I3 ) 2 Π2 = (α − Π2 ) (15.8.4) 1 I2 (I1 − I3 ) 2 and I3 (I1 − I2 ) 2 Π2 = (β − Π2 ), (15.8.5) 3 I2 (I1 − I3 ) 2 where α and β are positive constants given by aI2 (a − I3 )b2 aI2 (I1 − a)b2 α2 = and β 2 = . (15.8.6) I2 − I3 I1 − I2 By the deﬁnition of a, note that I1 ≥ a ≥ I3 . The endpoints of the interval [I1 , I3 ] are easy to deal with. If a = I1 , then Π2 = Π3 = 0 and the motion is a steady rotation about the Π-axis with body angular velocity ±b. Sim- ilarly, if a = I3 , then Π1 = Π2 = 0. So we can assume that I1 a I3 . With these expressions, the square of (15.8.1) becomes ˙ (Π2 )2 = a1 a3 (α2 − Π2 )(β 2 − Π2 ) 2 2 (15.8.7) that is, Π2 du t= (15.8.8) Π2 (0) a1 a3 (α2 − u2 )(β 2 − u2 ) which shows that Π2 , and hence Π1 , Π3 are elliptic functions of time. In case the quartic under the square root has double roots, that is, α = β, (15.8.8) can be integrated explicitly by means of elementary functions. By (15.8.6) if follows that ab2 I2 (I1 − I3 )(I2 − a) β 2 − α2 = . (I1 − I2 )(I2 − I3 ) Thus α = β if and only if a = I2 which in turn forces α = β = ab = Π and Π 2 = 2hI2 . Thus (15.8.7) becomes ˙ (Π2 )2 = a1 a3 ( Π 2 − Π2 )2 . 2 (15.8.9) ........................... 15 July 1998—18h02 ...........................

533. 15.8 The Analytical Solution of the Free Rigid Body Problem 519 If Π 2 = 2hI2 is satisﬁed, the intersection of the sphere of constant angular momentum Π with the elliptical energy surface corresponding to the value 2h consists of two great circles on the sphere going through the Π2 -axis in the planes a3 Π3 = ±Π1 . a1 In other words, the solution of (15.8.9) consists of four heteroclinic orbits and the values Π2 = ± Π . Equation (15.8.9) is solved by putting Π2 = Π tanh θ. Setting Π2 (0) = 0 for simplicity we get the four heteroclinic orbits a1 √ Π† (t) = ± Π sech (− a1 a3 Π t) , 1 −a2 √ Π† (t) = ± Π tanh (− a1 a3 Π t) , 2 (15.8.10) a3 √ Π† (t) = ± Π sech (− a1 a3 Π t) , 3 −a2 when a3 Π3 = Π1 a1 and Π− (t) = Π† (−t), Π− (t) = Π† (−t), Π− (t) = Π† (−t), 1 1 2 2 3 3 when a3 Π3 = −Π1 . a1 If α = β, then a = I2 , and the integration is performed with the aid of Jacobi’s elliptic functions (see Whittaker and Watson [1940], Chapter 22, or Lawden [1989]). For example, the elliptic function sn u with modulus k is given by 1 1 sn u = u − (1 + k 2 )u3 + (1 + 14k 2 + k 4 )u5 − . . . 3! 5! and its inverse is x 1 sn−1 x = dt, 0 ≤ x ≤ 1. 0 (1 − t2 )(1 − k 2 t2 ) Assuming I1 I2 a I3 or, equivalently, α β, the substitution of the elliptic function Π2 = α sn u in (15.8.8) with the modulus 1/2 (I1 − I2 )(a − I3 ) k = α/β = , (I1 − a)(I2 − I3 ) ........................... 15 July 1998—18h02 ...........................

534. 520 15. The Free Rigid Body gives u2 = ab2 (I1 − a)(I2 − I3 )/I1 I2 I3 = µ2 . We will need the identities ˙ d cn2 u = 1 − sn2 u, dn2 u = 1 − k 2 sn2 u, and sn x = cn x dn x. dx With initial condition Π2 (0) = 0, this gives Π2 = α sn (µt). (15.8.11) Thus Π2 varies between α and −α. Choosing the time direction appro- ˙ priately, we can assume without loss of generality that Π2 (0) 0. Note that Π1 vanishes when Π2 equals ±α by (15.8.4), but that Π2 attains its 3 maximal value I3 (I1 − I2 ) 2 I3 (I2 − a)ab2 (β − α2 ) = (15.8.12) I2 (I1 − I3 ) (I2 − I3 ) by (15.8.5). The minimal value of Π2 occurs when Π2 = 0, that is, it is 3 I3 (I1 − I2 ) 2 I3 (I1 − a)ab2 β = =: δ 2 , (15.8.13) I2 (I1 − I3 ) (I1 − I3 ) again by (15.8.5). Thus the sign of Π3 is constant throughout the motion. ˙ Let us assume it is positive. This hypothesis together with Π2 (0) 0 and a2 0 imply that Π1 (0) 0. Solving for Π1 and Π3 from (15.8.2) and (15.8.3) and remembering that Π1 (0) 0 gives Π1 (t) = −γ cn(µt), Π3 (t) = δ dn(µt), where δ is given by (15.8.13) and I1 (a − I3 )ab2 γ2 = . (15.8.14) (I1 − I3 ) Note that β α γ and, as usual, the values of γ and δ are taken to be positive. The solution of the Euler equations is therefore Π1 (t) = −γ cn(µt), Π2 (t) = α sn(µt), Π3 (t) = δ dn(µt), (15.8.15) with α, γ, δ given by (15.8.6), (15.8.13), (15.8.14). If κ denotes the period invariant of Jacobi’s elliptic functions then Π1 and Π2 have period 4κ/µ whereas Π3 has period 2κ/µ. Exercises 15.8-1. Continue this integration process and ﬁnd formulas for the attitude matrix A(t) as functions of time with A(0) = Identity and with given body angular momentum (or velocity). ........................... 15 July 1998—18h02 ...........................

535. 15.9 Rigid Body Stability 521 15.9 Rigid Body Stability Following the energy-Casimir method step by step (see the Introduction), we begin with the equations ˙ dΠ Π= = Π × Ω, (15.9.1) dt where Π, Ω ∈ R3 , Ω is the angular velocity, and Π is the angular momentum, both viewed in the body; the relation between Π and Ω is given by Πj = Ij Ωj , j = 1, 2, 3, where I = (I1 , I2 , I3 ) is the diagonalized moment of inertia tensor, I1 , I2 , I3 0. This system is Hamiltonian in the Lie– Poisson structure of R3 given by (15.3.8) and relative to the kinetic energy Hamiltonian 3 1 1 Π2 H(Π) = Π·Ω= i . (15.9.2) 2 2 i=1 Ii Recall from (15.3.12) that for a smooth function Φ : R → R, 1 2 CΦ (Π) = Φ Π (15.9.3) 2 is a Casimir function. 1 First Variation. We ﬁnd a Casimir function CΦ such that HCΦ := H + CΦ has a critical point at a given equilibrium point of (15.9.1). Such points occur when Π is parallel to Ω. We can assume without loss of generality, that Π and Ω point in the Ox-direction. After normalizing if necessary, we can assume that the equilibrium solution is Πe = (1, 0, 0). The derivative of 3 1 Π2 i 1 2 HCΦ (Π) = +Φ Π 2 i=1 Ii 2 is 1 DHCΦ (Π) · δΠ = Ω+Φ Π 2 Π · δΠ. (15.9.4) 2 This equals zero at Πe = (1, 0, 0), provided that 1 1 Φ =− . (15.9.5) 2 I1 ........................... 15 July 1998—18h02 ...........................

536. 522 15. The Free Rigid Body 2 Second Variation. Using (15.9.4), the second derivative of HCΦ at the equilibrium Πe = (1, 0, 0) is D2 HCΦ (Πe ) · (δΠ, δΠ) 1 1 = δΩ · δΠ + Φ Πe 2 δΠ 2 + (Πe · δΠ)2 Φ Πe 2 2 2 3 (δΠi )2 δΠ 2 1 = − +Φ (δΠ1 )2 i=1 Ii I1 2 1 1 1 1 1 = − (δΠ2 )2 + − (δΠ3 )2 + Φ (δΠ1 )2 . I2 I1 I3 I1 2 (15.9.6) 3 Definiteness. This quadratic form is positive-definite if and only if 1 Φ 0 (15.9.7) 2 and I1 I2 , I1 I3 . (15.9.8) Consequently, 2 1 1 Φ(x) = − x+ x− I1 2 satisfies (15.9.5) and makes the second derivative of HCΦ at (1, 0, 0) positive- definite, so stationary rotation around the shortest axis is (Liapunov ) stable. The quadratic form is negative-definite provided 1 Φ 0 (15.9.9) 2 and I1 I2 , I1 I3 . (15.9.10) It is obvious that we may find a function Φ satisfying the requirements 2 (15.9.5) and (15.9.9); for example, Φ(x) = −(1/I1 )x− x − 1 . This proves 2 that rotation around the long axis is (Liapunov ) stable. Finally, the quadratic form (15.9.6) is indefinite if I1 I2 , I3 I1 , (15.9.11) or the other way around. We cannot show by this method that rotation around the middle axis is unstable. We shall prove, by using a spectral ........................... 15 July 1998—18h02 ...........................

537. 15.9 Rigid Body Stability 523 analysis, that rotation about the middle axis is, in fact, unstable. Lineariz- ing (15.9.1) at Πe = (1, 0, 0) yields the linear constant coefficient system ˙ (δ Π) = δΠ × Ωe + Πe × δΩ I3 − I1 I1 − I2 = 0, δΠ3 , δΠ2 I3 I1 I1 I2   0 0 0  I3 − I1   0 0  = I3 I1  δΠ. (15.9.12)  I1 − I2  0 0 I1 I2 On the tangent space at Πe to the sphere of radius Πe = 1, the linear operator given by this linearized vector field has a matrix given by the lower right (2 × 2)-block whose eigenvalues are 1 ± √ (I1 − I2 )(I3 − I1 ). I1 I2 I3 Both of them are real by (15.9.11) and one is strictly positive. Thus Πe is spectrally unstable and thus is unstable. We summarize the results in the following theorem. Theorem 15.9.1 (Rigid Body Stability Theorem). In the motion of a free rigid body, rotation around the long and short axes is (Liapunov ) stable and around the middle axis is unstable. It is important to keep the Casimir functions as general as possible, because otherwise (15.9.5) and (15.9.9) could be contradictory. Had we simply chosen 2 1 1 Φ(x) = − x+ x− , I1 2 (15.9.5) would be verified, but (15.9.9) would not. It is only the choice of two different Casimirs that enables us to prove the two stability results, even though the level surfaces of these Casimirs are the same. Remarks. 1. As we have seen, rotations about the intermediate axis are unstable and this is even for the linearized equations. The unstable homoclinic orbit that connect the two unstable points have interesting features. Not only are they interesting because of the chaotic solutions via the Poincar´-Melnikov e method that can be obtained in various perturbed systems (see Holmes and Marsden [1983], Wiggins [1988], and references therein), but already, the ........................... 15 July 1998—18h02 ...........................

538. 524 15. The Free Rigid Body orbit itself is interesting since a rigid body tossed about its middle axis will undergo an interesting half twist when the opposite saddle point is reached, even though the rotation axis has returned to where it was. The reader can easily perform the experiment; see Ashbaugh, Chicone, and Cushman [1990] and Montgomery [1991a] for more information. 2. The same stability theorem can also be proved by working with the second derivative along a coadjoint orbit in R3 ; that is, a two-sphere; see Arnold [1966a]. This coadjoint orbit method also suggests instability of rotation around the intermediate axis. 3. Dynamic stability on the Π-sphere has been shown. What about the stability of the dynamically rigid body we “see”? This can be deduced from what we have done. Probably the best approach though is to use the relation between the reduced and unreduced dynamics; see Simo, Lewis, and Marsden [1991] and Lewis [1992] for more information. 4. When the body angular momentum undergoes a periodic motion, the actual motion of the rigid body in space is not periodic. In the introduction we described the associated geometric phase. 5. See Lewis and Simo [1990] and Simo, Lewis, and Marsden [1991] for related work on deformable elastic bodies (pseudo-rigid bodies). Exercises 15.9-1. Let B be a given fixed vector in R3 and let M evolve by M = ˙ M × B. Show that this evolution is Hamiltonian. Determine the equilibria and their stability. 15.9-2. Consider the following modification of the Euler equations: ˙ Π = Π × Ω + αΠ × (Π × Ω), where α is a positive constant. Show that, 2 (a) The spheres Π are preserved. (b) Energy is strictly decreasing except at equilibria. (c) The equations can be written in the form ˙ F = {F, H}rb + {F, H}sym , where the first bracket is the usual rigid body bracket and the second is the symmetric bracket {F, K}sym = α(Π × F ) · (Π × K). ........................... 15 July 1998—18h02 ...........................

539. 15.10 Heavy Top Stability 525 15.10 Heavy Top Stability The heavy top equations are dΠ = Π × Ω + M glΓ × χ, (15.10.1) dt dΓ = Γ × Ω, (15.10.2) dt where Π, Γ, χ ∈ R3 . Here Π and Ω are the angular momentum and angular velocity in the body, Πi = Ii Ωi , Ii 0, i = 1, 2, 3, with I = (I1 , I2 , I3 ) the moment of inertia tensor. The vector Γ represents the motion of the unit vector along the Oz-axis as seen from the body, and the constant vector χ is the unit vector along the line segment of length l connecting the ﬁxed point to the center mass of the body; M is the total mass of the body, and g is the strength of the gravitational acceleration, which is along Oz pointing down. This system is Hamiltonian in the Lie–Poisson structure of R3 ×R3 given in the Introduction relative to the heavy top Hamiltonian 1 H(Π, Γ) = Π · Ω + M glΓ · χ. (15.10.3) 2 The Poisson structure (with Π = 1 imposed) foreshadows that of T ∗ SO(3)/S 1 , where S 1 acts by rotation about the axis of gravity. The fact that one gets the Lie–Poisson bracket for a semi-direct product Lie algebra is a special case of the general theory of reduction and semi-direct products (Marsden, Ratiu and Weinstein [1984a,b]) The functions Π · Γ and Γ 2 are Casimir functions, as is C(Π, Γ) = Φ(Π · Γ, Γ 2 ), (15.10.4) where Φ is any smooth function from R to R. 2 We shall be concerned here with the Lagrange top. This is a heavy top for which I1 = I2 , that is, it is symmetric, and the center of mass lies on the axis of symmetry in the body, that is, χ = (0, 0, 1). This assumption simpliﬁes the equations of motion (15.10.1) to ˙ I2 − I3 Π1 = Π2 Π3 + M glΓ2 , I2 I3 ˙ I3 − I1 Π2 = Π1 Π3 − M glΓ1 , I1 I3 ˙ I1 − I2 Π3 = Π1 Π2 . I1 I2 ˙ Since I1 = I2 , we have Π3 = 0; thus Π3 and hence any function ϕ(Π3 ) of Π3 is conserved. ........................... 15 July 1998—18h02 ...........................

540. 526 15. The Free Rigid Body 1 First Variation. We shall study the equilibrium solution Πe = (0, 0, Π0 ), 3 Γe = (0, 0, 1), where Π0 3 = 0, which represents the spinning of a symmetric top in its upright position. To begin, we consider conserved quantities of the form HΦ,ϕ = H + Φ(Π · Γ, Γ 2 ) + ϕ(Π3 ) and which have a critical point at the equilibrium. The first derivative of HΦ,ϕ is given by ˙ DHΦ,ϕ (Π, Γ) · (δΠ, δΓ) = (Ω + Φ(Π · Γ, Γ 2 )Γ) · δΠ ˙ + [M glχ + Φ(Π · Γ, Γ 2 )Π + 2Φ (Π · Γ, Γ 2 )Γ] · δΓ + ϕ (Π3 )δΠ3 , ˙ where Φ = ∂Φ/∂(Π · Γ) and Φ = ∂Φ/∂( Γ 2 ). At the equilibrium solution (Πe , Γe ) the first derivative of HΦ,ϕ vanishes, provided that Π0 3 ˙ 3 + Φ(Π0 , 1) + ϕ (Π0 ) = 0 3 I3 and that ˙ 3 M gl + Φ(Π0 , 1)Π0 + 2Φ (Π0 , 1) = 0; 3 3 the remaining equations, involving indices 1 and 2, are trivially verified. ˙ Solving for Φ(Π0 , 1) and Φ (Π0 , 1) we get the conditions 3 3 ˙ 1 ϕ (Π0 ) Φ(Π0 , 1) = − 3 + 3 Π0 , 3 (15.10.5) I3 Π0 3 1 1 ϕ (Π0 ) 1 Φ (Π0 , 1) = 3 + 0 3 (Π0 )2 − M gl. 3 (15.10.6) 2 I3 Π3 2 2 Second Variation. We shall check for definiteness of the second variation of HΦ,ϕ at the equilibrium point (Πe , Γe ). To simplify the notation we shall set a = ϕ (Π0 ), b = 4Φ (Π0 , 1), ¨ c = Φ(Π0 , 1), ˙ d = 2Φ (Π0 , 1). 3 3 3 3 With this notation, the matrix of the second derivative at (Πe , Γe ) is  ˙  1/I1 0 0 Φ(Π0 , 1) 3 0 0  0 1/I1 0 0 ˙ Φ(Π0 , 1) 0   3   0 0 (1/I3 ) + a + c 0 0 a36   ,  Φ(Π0 , 1) ˙ 0 0 2Φ (Π0 , 1) 0 0   3 3   0 ˙ Φ(Π0 , 1) 0 0 2Φ (Π0 , 1) 0  3 3 0 0 a36 0 0 a66 (15.10.7) where ˙ 3 a36 = Φ(Π0 , 1) + Π0 c + d, 3 a66 = 2Φ (Π0 , 1) + b + (Π0 )2 c + Π0 d. 3 3 3 ........................... 15 July 1998—18h02 ...........................

541. 15.10 Heavy Top Stability 527 3 Definiteness. The computations for this part will be done using the following formula from linear algebra. If A B M= C D is a (p + q) × (p + q) matrix and if the (p × p)-matrix A is invertible, then det M = det A det(D − CA−1 B). If the quadratric form given by (15.10.7) is definite, it must be positive- definite since the (1, 1)-entry is positive. Recalling that I1 = I2 , the six principal determinants have the following values: 1 1 1 1 , 2, 2 +a+c , I1 I1 I1 I3 1 1 2 ˙ +a+c Φ (Π0 , 1) − Φ(Π0 , 1)2 , 3 3 I1 I3 I1 2 2 ˙ 3 1 Φ (Π0 , 1) − Φ(Π0 , 1)2 3 +a+c , I1 I3 and 2 2 ˙ 3 1 Φ (Π0 , 1) − Φ(Π0 , 1)2 3 a66 + a + c − a2 . 36 I1 I3 Consequently, the quadratic form given by (15.10.7) is positive-definite, if and only if 1 + a + c 0, (15.10.8) I3 2 ˙ 3 Φ (Π0 , 1) − Φ(Π0 , 1)2 0, 3 (15.10.9) I1 and 1 2 a66 ˙ + a + c − Φ Π0 , 1 + Π0 c + d 3 3 0. (15.10.10) I3 Conditions (15.10.8) and (15.10.10) can always be satisfied if we choose the numbers a, b, c, and d appropriately; for example, a = c = d = 0 and b sufficiently large and positive. Thus, the determining condition for stability is (15.10.9). By (15.10.5) and (15.10.6), this becomes 2 1 1 ϕ (Π0 ) 1 ϕ (Π0 ) + 3 (Π0 )2 − M gl − 3 + 3 (Π0 )2 0. 3 I1 I3 Π0 3 I3 Π0 3 (15.10.11) ........................... 15 July 1998—18h02 ...........................

542. 528 15. The Free Rigid Body We can choose ϕ (Π0 ) so that 3 1 ϕ (Π0 ) 3 + =e I3 Π0 3 has any value we wish. The left side of (15.10.11) is a quadratic polynomial in e, whose leading coefficient is negative. In order for this to be positive for some e, it is necessary and sufficient for the discriminant (Π0 )4 4(Π0 )2 M gl 3 2 − 3 I1 I1 to be positive; that is, (Π0 )2 4M glI1 3 which is the classical stability condition for a fast top. We have proved the first part of the following: Theorem 15.10.1 (Heavy Top Stability Theorem). An upright spinning Lagrange top is stable provided that the angular velocity is strictly √ larger than 2 M glI1 /I3 . It is unstable if the angular velocity is smaller than this value. The second part of the theorem is proved, as in §15.9, by a spectral analysis of the linearized equations, namely ˙ (δ Π) = δΠ × Ω + Πe × δΩ + M glδΓ × χ, (15.10.12) ˙ (δ Γ) = δΓ × Ω + Γe × δΩ, (15.10.13) on the tangent space to the coadjoint orbit in se(3)∗ through (Πe , Γe ) given by {(δΠ, δΓ) ∈ R3 × R3 | δΠ · Γe + Πe · δΓ = 0 and δΓ · Γe = 0} ∼ = {(δΠ1 , δΠ2 , δΓ1 , δΓ2 )} = R4 . (15.10.14) The matrix of the linearized system of equations on this space is computed to be   Π0 I1 − I3 3  0 0 M gl   I3 I1   Π0 I1 − I3   − 3 0 −M gl 0   I3 I1   . (15.10.15)  1 Π0   0 − 0 3   I1 I3   1 Π 0  0 − 3 0 I1 I3 ........................... 15 July 1998—18h02 ...........................

543. 15.11 The Rigid Body and the Pendulum 529 The matrix (15.10.15) has characteristic polynomial 2 1 Π0 2 (I1 + (I1 − I3 ) ) − 2M glI1 λ2 3 λ4 + 2 2 I1 I3 2 2 1 Π0 2 (I1 − I3 ) 3 + + M gl , (15.10.16) I1 I3 whose discriminant as a quadratic polynomial in λ2 is 2 2 1 Π0 Π0 4 (2I1 − I3 ) − 4M glI1 2 3 2 3 I3 . I1 I3 I3 This discriminant is negative if and only if Π0 2 M glI1 . 3 Under this condition the four roots of the characteristic polynomial are all ¯ ¯ distinct and equal to λ0 , λ0 , −λ0 , −λ0 for some λ0 ∈ C, where Re λ0 = 0 and Im λ0 = 0. Thus, at least two of these roots have real part strictly larger than zero thereby showing that (Πe , Γe ) is spectrally unstable and hence unstable. When I2 = I1 + for small , the conserved quantity ϕ(Π3 ) is no longer available. In this case, a suﬃciently fast top is still linearly stable, and nonlinear stability can be assessed by KAM theory. Other regions of phase space are known to possess chaotic dynamics in this case (Holmes and Marsden [1983]). For more information on stability and bifurcation in the heavy top, we refer to Lewis, Ratiu, Simo, and Marsden [1992]. Exercises 15.10-1. ˜ (a) Show that H(Π, Γ) = H(Π, Γ)+ Γ 2 /2, where H is given by (15.10.3), generates the same equations of motion (15.10.1) and (15.10.2). ˜ (b) Taking the Legendre transform of H, show that the equations can be written in Euler–Poincar´ form. e 15.11 The Rigid Body and the Pendulum This section, following Holm and Marsden [1991], shows how the rigid body and the pendulum are linked. Euler’s equations are expressible in vector form as ........................... 15 July 1998—18h02 ...........................

544. 530 15. The Free Rigid Body d Π= L× H, (15.11.1) dt where H is the energy, Π21 Π2 Π2 H= + 2 + 3, (15.11.2) 2I1 2I2 2I3 ∂H ∂H ∂H Π1 Π2 Π3 H= , , = , , , (15.11.3) ∂Π1 ∂Π2 ∂Π3 I1 I2 I3 is the gradient of H and L is the square of the body angular momentum, 1 2 L= Π + Π2 + Π2 . (15.11.4) 2 1 2 3 Since both H and L are conserved, the rigid body motion itself takes place, as we know, along the intersections of the level surfaces of the energy (ellipsoids) and the angular momentum (spheres) in R3 . The centers of the energy ellipsoids and the angular momentum spheres coincide. This, along with the (Z2 )3 symmetry of the energy ellipsoids, implies that the two sets of level surfaces in R3 develop collinear gradients (for example, tangencies) at pairs of points which are diametrically opposite on an angular momentum sphere. At these points, collinearity of the gradients of H and L implies stationary rotations, that is, equilibria. Euler’s equations for the rigid body may also be written as d Π= N× K, (15.11.5) dt where K and N are linear combinations of energy and angular momentum of the form N a b H = , (15.11.6) K c d L with real constants a, b, c, and d satisfying ad − bc = 1. To see this recall that 1 c 1 c 1 c K= + d Π2 + 1 + d Π2 + 2 + d Π2 . 3 2 I1 2 I2 2 I3 Thus, if I1 = I2 = I3 , the choice c = −dI1 yields K = 0, and so equa- ˙ tion (15.11.5) becomes Π = 0 for any choice of N , which is precisely the equation Π˙ = Π × Ω, for I1 = I2 = I3 . If I1 = I2 = I3 the choice c = −dI2 , d = 0, yields d I2 K= 1− Π2 . 1 2 I1 ........................... 15 July 1998—18h02 ...........................

545. 15.11 The Rigid Body and the Pendulum 531 If one now takes I1 N= Π2 + Π2 , 2I2 d 2 3 ˙ then equation (15.11.5) becomes the rigid body equation Π = Π × Ω. Finally, if I1 I2 I3 , the choice 1 I1 I3 I3 c = 1, d=− , a=− 0, and b = 0 (15.11.7) I3 I3 − I1 I3 − I1 gives 1 1 1 1 1 1 K= − Π2 + 1 − Π2 2 (15.11.8) 2 I1 I3 2 I2 I3 and I3 (I2 − I1 ) 2 1 2 N= Π + Π . (15.11.9) 2I2 (I3 − I1 ) 2 2 3 ˙ Then equations (15.11.5) are the rigid body equation Π = Π × Ω With this choice, the orbits for Euler’s equations for rigid body dynamics are realized as motion along the intersections of two, orthogonally oriented, elliptic cylinders, one elliptic cylinder being a level surface of K, with its translation axis along Π3 (where K = 0 ), and the other a level surface of N , with its translation axis along Π1 (where N = 0). For a general choice of K and N , equilibria occur at points where the gradients of K and N are collinear. This can occur at points where the level sets are tangent (and the gradients both are nonzero), or at points where one of the gradients vanishes. In the elliptic cylinder case above, these two cases are points where the elliptic cylinders are tangent, and at points where the axis of one cylinder punctures normally through the surface of the other. The elliptic cylinders are tangent at one Z2 -symmetric pair of points along the Π2 axis, and the elliptic cylinders have normal axial punctures at two other Z2 -symmetric pairs of points along the Π1 and Π3 axes. Let us pursue the elliptic cylinders point of view further. We now change variables in the rigid body equations within a level surface of K. To sim- 2 plify notation, we ﬁrst deﬁne the three positive constants ki , i = 1, 2, 3, by setting in (15.11.8) and (15.11.9) Π21 Π22 Π22 1 2 K= 2 + 2k 2 and N = 2 + 2 Π3 . (15.11.10) 2k1 2 2k3 For 1 1 1 1 1 1 1 I3 (I2 − I1 ) 2 = I − I , k1 2 = I − I , k2 2 = I (I − I ) . k3 (15.11.11) 1 3 2 3 2 3 1 ........................... 15 July 1998—18h02 ...........................

546. 532 15. The Free Rigid Body √ On the surface K = constant, and setting r = 2K = constant, deﬁne new variables θ and p by Π1 = k1 r cos θ, Π2 = k2 r sin θ, Π3 = p. (15.11.12) In terms of these variables, the constants of the motion become 2 1 2 1 2 k2 2 K= r and N = p + 2r sin2 θ. (15.11.13) 2 2 2k3 From Exercise 1.3-2 it follows that {F1 , F2 }K = − K · ( F1 × F2 ). (15.11.14) is a Poisson bracket on R3 having K as a Casimir function. One can now verify directly that the symplectic structure on the leaf K = constant is given by the following Poisson bracket on this elliptic cylinder (see Exer- cise 15.11-1): 1 ∂F ∂G ∂F ∂G {F, G}EllipCyl = − . (15.11.15) k1 k2 ∂p ∂θ ∂θ ∂p In particular, 1 {p, θ}EllipCyl = . (15.11.16) k1 k2 The restriction of the Hamiltonian H to the elliptic cylinder K = constant is by (15.11.3) 2 k1 K 1 1 2 I3 (I2 − I1 ) 2 H= + p + r2 sin2 θ I1 I3 2 2(I3 − I2 )(I3 − I1 ) k2 K 1 = 1 + N, I1 I3 that is, N/I3 can be taken as the Hamiltonian on this symplectic leaf. Note that N/I3 has the form of kinetic plus potential energy. The equations of motion are thus given by d N 1 ∂N 1 θ= θ, = =− p, (15.11.17) dt I3 EllipCyl k1 k2 I3 ∂p k1 k2 I3 2 d N 1 ∂N 1 k2 2 p= p, = = 2 r sin θ cos θ. (15.11.18) dt I3 EllipCyl k1 k2 I3 ∂θ k1 k2 I3 k3 Combining these equations of motion gives d2 θ r2 = − 2 2 2 sin 2θ, (15.11.19) dt2 2k1 k3 I3 ........................... 15 July 1998—18h02 ...........................

547. 15.11 The Rigid Body and the Pendulum 533 or, in terms of the original rigid body parameters, d2 K 1 1 2 θ=− 2 − sin 2θ. (15.11.20) dt I3 I1 I2 Thus, we have proved Proposition 15.11.1. Rigid body motion reduces to pendulum motion on level surfaces of K. Another way of saying this is as follows: regard rigid body angular momentum space as the union of the level surfaces of K, so the dynamics of the rigid body is recovered by looking at the dynamics on each of these level surfaces. On each level surface, the dynamics is equivalent to a simple pendulum. In this sense, we have proved: Corollary 15.11.2. The dynamics of a rigid body in three-dimensional body angular momentum space is a union of two-dimensional simple pen- dula phase portraits. By restricting to a nonzero level surface of K, the pair of rigid body equilibria along the Π3 axis are excluded. (This pair of equilibria can be included by permuting the indices of the moments of inertia.) The other two pairs of equilibria, along the Π1 and Π2 axes, lie in the p = 0 plane at θ = 0, π/2, π, and 3π/2. Since K is positive, the stability of each equilibrium point is determined by the relative sizes of the principal moments of inertia, which aﬀect the overall sign of the right-hand side of the pendulum equation. The well-known results about stability of equilibrium rotations along the least and greatest principal axes, and instability around the intermediate axis, are immediately recovered from this overall sign, combined with the stability properties of the pendulum equilibria. For K 0 and I1 I2 I3 , this overall sign is negative, so the equilibria at θ = 0 and π (along the Π1 axis) are stable, while those at θ = π/2 and 3π/2 (along the Π2 axis) are unstable. The factor of 2 in the argument of the sine in the pendulum equation is explained by the Z2 symmetry of the level surfaces of K (or, just as well, by their invariance under θ → θ + π). Under this discrete symmetry operation, the equilibria at θ = 0 and π/2 exchange with their counterparts at θ = π and 3π/2, respectively, while the elliptical level surface of K is left invariant. By construction, the Hamiltonian N/I3 in the reduced variables θ and p is also invariant under this discrete symmetry. The rigid body can, correspondingly, be regarded as a left invariant system on the group O(K) or SE(2). The special case of SE(2) is the one in which the orbits are cotangent bundles. The fact that one gets a cotangent bundle in this situation is a special case of the cotangent bundle reduction theorem using the semidirect product reduction theorem; see Marsden, Ratiu, and Weinstein [1984a,b]. For the Euclidean group it says that the coadjoint orbits of the Euclidean group of the plane are given by reducing ........................... 15 July 1998—18h02 ...........................

548. 534 15. The Free Rigid Body the cotangent bundle of the rotation group of the plane by the trivial group, giving the cotangent bundle of a circle with its canonical symplectic structure up to a factor. This is the abstract explanation of why, in the elliptic cylinder case above, the variables θ and p were, up to a factor, canonically conjugate. This general theory is also consistent with the fact that the Hamiltonian N/I3 is of the form kinetic plus potential energy. In fact, in the cotangent bundle reduction theorem, one always gets a Hamiltonian of this form, with the potential being changed by the addition of an amendment to give the amended potential. In the case of the pendulum equation, the original Hamiltonian is purely kinetic energy and so the potential term in N/I3 , namely (k2 r2 /2k3 I3 ) sin2 θ, is entirely amendment. 2 2 Putting the above discussion together with Exercises 14.7-1 and 14.7-2, one gets Theorem 15.11.3. Euler’s equations for a free rigid body are Lie–Poisson with the Hamiltonian N for the Lie algebra R3 where the underlying Lie K group is the orthogonal group of K if the quadratic form is nondegenerate, and is the Euclidean group of the plane if K has signature (+, +, 0). In particular, all the groups SO(3), SO(2, 1), and SE(2) occur as the parameters a, b, c, and d are varied. (If the body is a Lagrange body, then the Heisenberg group occurs as well.) The same richness of Hamiltonian structure was found in the Maxwell– Bloch system in David and Holm [1992] (see also David, Holm, and Tratnick [1990]). As in the case of the rigid body, the R3 motion for the Maxwell– Bloch system may also be realized as motion along the intersections of two orthogonally oriented cylinders. However, in this case, one cylinder is parabolic in cross section, while the other is circular. Upon passing to parabolic cylindrical coordinates, the Maxwell–Bloch system reduces to the ideal Duﬃng equation, while in circular cylindrical coordinates, the pendulum equation results. The SL(2, R) matrix transformation in the Maxwell– Bloch case provides a parametrized array of (oﬀset) ellipsoids, hyperboloids, and cylinders, along whose intersections the R3 motion takes place. Exercises 15.11-1. Consider the Poisson bracket on R3 given by {F1 , F2 }K (Π) = − K (Π) · ( F1 (Π) × ( F2 (Π)) with Π21 Π2 K(Π) = 2 + 2.2 2k1 2k2 Verify that the Poisson bracket on the two-dimensional leaves of this bracket given by K = constant has the expression 1 {θ, p}Ellip Cyl = − , k1 k2 ........................... 15 July 1998—18h02 ...........................

549. 15.11 The Rigid Body and the Pendulum 535 where p = Π3 and θ = tan−1 (k1 Π2 /k2 Π1 ). What is the symplectic form on these leaves? ........................... 15 July 1998—18h02 ...........................

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