Hamilton application
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Central problem in mechanics is describing a system's mechanical state and how it evolves over time. Three formulations include Galileo/Newton using coordinates and velocities, Lagrange using generalized coordinates and velocities, and Hamilton using positions and momenta in phase space. Time homogeneity leads to conservation of energy from the Lagrangian not explicitly depending on time. Space homogeneity leads to conservation of momentum from the Lagrangian being independent of position coordinates. Noether's theorem links symmetries like time and space homogeneity to conserved quantities like energy and momentum.
![Laplace Runge Lenz Vector is
constant for a strict 1/r potential.
Reference: Goldstein’s ‘Classical Mechanics’, Section 9, Chapter 3.
2 2 21
( ) ( )
2
( )
L T V m V
k
V
ρ ρ ϕ ρ
ρ
ρ
= − = + −
= −
& &
ˆA p L mkeρ= × −
ur ur ur
2
0,
one requires
dp
ˆ
dt
DYNAM ICAL
SYMM ETRY
dA
For
dt
k
eρ
ρ
=
= −
uur
r
For (angular momentum vector)
to be conserved,
any central force would do.
[Geometrical Symmetry]
L
ur
LRL figure from http://en.wikipedia.org/wiki/Laplace-Runge-Lenz_vector
PCD-08
p
ur L×
ur
ˆeρ
A
ur
p L×
ur ur
ˆmkeρ−](https://image.slidesharecdn.com/hamiltonapplication-170206182020/85/Hamilton-application-13-320.jpg)
Hamilton application
- 1. Central problem in ‘Mechanics’: How is the ‘mechanical state’ of a system described, and how does this ‘state’ evolve with time? Formulations due to Galileo/Newton, Lagrange and HamiltonPCD-08
- 2. coordinate , velocity is related to momentum dq q q dt q p ⎛ ⎞ =⎜ ⎟ ⎝ ⎠ & & Equation of motion: relation between , andq q q& && Causality and determinism, Newton’s second law state of the system in Quantum mechanics: ‘Position/Momentum uncertainty’ p q . (q,p) Point in ‘phase space’ specifies the ‘state’ of the system. We need dq/dt and dp/dt‘Mechanics’ by L&L, III Edition PCD-08
- 3. Homogeneity with respect to time/space translations and isotropy of space, inertial frame : The laws of mechanics are the same in an infinity of inertial reference frames moving, relative to one another, uniformly in a straight line. If the position of particle is given by the vector ( ) in one frame of reference, and by (t) in another frame of reference moving at a constant velocity v with respect the previous one, then r t r r ′ r ur r '( ) ( ) ; is 'absolute' in the two frames: t r t t TIME t t = + ′= ur rr v “GALILEAN PRINCIPLE OF RELATIVITY” PCD-08
- 4. In an inertial frame, Time is homogeneous Space is homogenous and isotropic Every mechanical system is characterized by a function ( , , ),the Lagrangian of the systemL q q t& 2 1 ( , , ) . t t S L q q t dtaction = ∫ & Mechanical state of a system 'evolves' (along a 'world line') in such a way that ' ', is an extremum HamiltonHamilton’’s principles principle ‘‘principle of leastprinciple of least (rather,(rather, extremumextremum)) actionaction’’ PCD-08
- 5. 0 S S Sδ = would be an extremum when the variation in is zero; i.e. 2 1 ( , , ) t t S L q q t dtaction = ∫ & motion takes place in such a way that ' ', is an extremum 2 2 1 1 ( , , ) ( , , ) 0 t t t t S L q q q q t dt L q q t dtδ δ δ= + + − =∫ ∫& & & ( ) 2 2 1 1 . . 0 t t t t L L L L d i e S q q dt q q dt q q q q dt δ δ δ δ δ ⎧ ⎫ ⎧ ⎫∂ ∂ ∂ ∂ = = + = +⎨ ⎬ ⎨ ⎬ ∂ ∂ ∂ ∂⎩ ⎭ ⎩ ⎭ ∫ ∫& & & ( )2 2 1 1 22 2 1 11 . . 0 0 t t t t tt t t tt d qL L i e S q dt dt q q dt L L d L q dt q q dt q q dt q δ δ δ δ δ δ ⎧ ⎫⎧ ⎫∂ ∂ = = +⎨ ⎬ ⎨ ⎬ ∂ ∂⎩ ⎭ ⎩ ⎭ ⎧ ⎫⎧ ⎫ ⎡ ⎤ ⎛ ⎞∂ ∂ ∂ = + −⎨ ⎬ ⎨ ⎬⎜ ⎟⎢ ⎥∂ ∂ ∂⎩ ⎭ ⎣ ⎦ ⎝ ⎠⎩ ⎭ ∫ ∫ ∫ ∫ & & & Integration by parts PCD-08
- 6. 2 2 11 . . 0 t t tt L L d L i e q qdt q q dt q δ δ ⎧ ⎫⎡ ⎤ ⎛ ⎞∂ ∂ ∂ = + −⎨ ⎬⎜ ⎟⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎝ ⎠⎩ ⎭ ∫& & ( ) ( )1 2Now, = 0, and is an arbitrary variation. Hence, 0 ' q t q t q L d L Lagrange s Equation q dt q δ δ δ= ⎛ ⎞∂ ∂ − =⎜ ⎟ ∂ ∂⎝ ⎠& Lagrange’s equation of motion 2 1 2 2 ( , , ) ( ) ( ) ( ) 2 - , so, . Also, , the momentum L d L q dt q L q q t f q f q m q V q T V L V L F mq p q q q ⎛ ⎞∂ ∂ = ⎜ ⎟ ∂ ∂⎝ ⎠ = + = − = ∂ ∂ ∂ = − = = = ∂ ∂ ∂ & & & & & & i.e., : in 3D: ( ) ' dp dp F F V q Newton s II Law dt dt = = = −∇ ⇔ r r r Homogeneity & Isotropy of space ⇒L can depend only quadratically on the velocity. PCD-08
- 7. Law of conservation of energy arises from the homogeneity of time. dL d (1) 0 + q + dt dt d (2) 0 q dt (1) (2) d - 0 dt - i i i i L L L L q q q q q q q L q Equations and L q L q L q L q ∂ ∂ ∂ ∂ = = = ∂ ∂ ∂ ∂ ⎡ ⎤∂ = ⎢ ⎥∂⎣ ⎦ ⇒ ⎡ ⎤∂ =⎢ ⎥ ∂⎣ ⎦ ⎡ ⎤∂ ⎢ ⎥ ∂⎣ ⎦ ∑ ∑ && && && & & & & & & & & & is a CONSTANT: - -i i i i ENERGY Hamiltonian L H q L q p L q ⎡ ⎤∂ ⎡ ⎤= =⎢ ⎥ ⎣ ⎦∂⎣ ⎦ ∑ ∑& & & Summation over i: degrees of freedom USING LAGRANGE’s EQUATION PCD-08 Time is homogeneous: Lagrangian of a closed system does not depend explicitly on time.
- 8. Time is homogeneous: Lagrangian of a closed system does not depend explicitly on time. Law of conservation of energy arises from the homogeneity of time. dL d (1) 0 + q + dt dt d (2) 0 q dt (1) (2) d - 0 dt - i i i i L L L L q q q q q q q L q Equations and L q L q L q L q ∂ ∂ ∂ ∂ = = = ∂ ∂ ∂ ∂ ⎡ ⎤∂ = ⎢ ⎥∂⎣ ⎦ ⇒ ⎡ ⎤∂ =⎢ ⎥ ∂⎣ ⎦ ⎡ ⎤∂ ⎢ ⎥ ∂⎣ ⎦ ∑ ∑ && && && & & & & & & & & & is a CONSTANT: - -i i i i ENERGY Hamiltonian L H q L q p L q ⎡ ⎤∂ ⎡ ⎤= =⎢ ⎥ ⎣ ⎦∂⎣ ⎦ ∑ ∑& & & USING LAGRANGE’s EQUATION Summation over i: degrees of freedom PCD-08
- 9. Hamiltonian (Hamilton’s Principal Function) of a system k k k k k k k k k k k k kk k k k k k k k k k k k k k H q p L L L dH p dq q dp dq dq q q L q dp dq q q dp p dq = − ∂ ∂ = + − − ∂ ∂ ∂ = − ∂ = − ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ & & & & & & & & k k k kk k k k k k H H dp dq p q H H q p p q ∂ ∂ + ∂ ∂ ∂ ∂ ∀ = = − ∂ ∂ ∑ ∑ & & k kBut, H=H(p ,q ) so dH = Hence k: and Hamilton’s equations of motion PCD-08
- 10. since 0, this means . . is conserved. . ., is independent of time, is a constant of motion L d L L i e p q dt q q i e ⎛ ⎞∂ ∂ ∂ − = =⎜ ⎟ ∂ ∂ ∂⎝ ⎠& & In an inertial frame, Time is homogeneous; Space is homogenous and isotropic Law of conservation of momentum, arises from the homogeneity of space. the condition for homogeneity of space : ( , , ) 0 . ., 0 which implies 0 where , , L x y z L L L i e L x y z x y z L q x y z q δ δ δ δ δ = ∂ ∂ ∂ = + + = ∂ ∂ ∂ ∂ = = ∂ PCD-08
- 11. NOETHER’s THEOREM: Emmy Noether 1882 to 1935 SYMMETRY CONSERVATION PRINCIPLE Homogeneity of time Energy Homogeneity of space Linear Momentum Isotropy of Space Angular momentum CPT Theorem: Standard ModelPCD-08
- 12. DYNAMICAL SYMMETRY, (‘accidental’ symmetry) rather than GEOMETRICAL SYMMETRY Laplace Runge Lenz Vector – constant for a strict 1/r potential Force: -1/r2 Why is the ellipse in the Kepler problem fixed? What ‘else’ is conserved? PCD-08
- 13. Laplace Runge Lenz Vector is constant for a strict 1/r potential. Reference: Goldstein’s ‘Classical Mechanics’, Section 9, Chapter 3. 2 2 21 ( ) ( ) 2 ( ) L T V m V k V ρ ρ ϕ ρ ρ ρ = − = + − = − & & ˆA p L mkeρ= × − ur ur ur 2 0, one requires dp ˆ dt DYNAM ICAL SYMM ETRY dA For dt k eρ ρ = = − uur r For (angular momentum vector) to be conserved, any central force would do. [Geometrical Symmetry] L ur LRL figure from http://en.wikipedia.org/wiki/Laplace-Runge-Lenz_vector PCD-08 p ur L× ur ˆeρ A ur p L× ur ur ˆmkeρ−
- 14. Unit 5 (Sept. 1-5): Kepler Problem. Laplace-Runge-Lenz vector, ‘Dynamical’ symmetry. Conservation principle ↔ Symmetry relation. U5L1: Kepler Problem. Laplace-Runge-Lenz vector, ‘Dynamical’ symmetry. U5L2: Conservation principle ↔ Symmetry relation. T5: on 8th September, Monday. T4: 1st September, Monday Pierre-Simon Laplace 1749 - 1827 Carl David Tolmé Runge 1856 - 1927 Wilhelm Lenz 1888 -1957 Symmetry of the H atom: ‘old’ quantum theory. En ~ n-2 PCD-08
- 15. A simple illustration: one-dimensional motion along Cartesian x-axis – this example highlights ‘additivity’ of the action integral as limit of a sum. Hanc, Taylor and Tuleja, Am. J. Phys., 72(4), 2004 PCD-08
- 16. Principle of least action: Hamilton’s principle “actionaction” as an additive property: L=L(x,v) PCD-08
- 17. Principle of least action: Hamilton’s principle Leads to LAGRANGE’s Eq. To first order, the first term is the average value ∂L/∂x on the two segments A and B. In the limit ∆t→0, this term approaches the value of the partial derivative of L at x. In the same limit, the second term is the time derivative of the partial derivative of the Lagrangian with respect to velocity d(∂L/∂v)/dt. PCD-08
- 18. special case: L is NOT a function of x : “ignorable” or “cyclic” coordinate ( ) 0 ( ) L L d L dt L m p = ∂⎧ ⎫ =⎨ ⎬ ∂⎩ ⎭ ∂ = ∂ Then the Lagrangian and the Lagrange's equation reduces to which means is a contant of motion. v v v = v ( )L V x x x L m p force linear momentum ∂ ∂ = − = ∂ ∂ ∂ = = = ∂ meaning and physical significane of the two terms? note that the and thev v Newton’s Second Law! Thus translational symmetry ( i.e. L being independent of x ) leads to the conservation of linear momentum!PCD-08
- 19. Hanc, Taylor and Tuleja, Am. J. Phys., 72(4), 2004 HOMOGENEITY WITH RESPECT TO “TIME” PCD-08
- 20. i.e. is a constant of motion L L ∂⎧ ⎫ −⎨ ⎬ ∂⎩ ⎭ v v 2 2 21 1 ( ) ( ) 2 2 L L m m V x m V x ∂⎧ ⎫ ⎧ ⎫ − = − − = +⎨ ⎬ ⎨ ⎬ ∂⎩ ⎭ ⎩ ⎭ but v v v v v The total energy of the system is a constant of motion ( is conserved) Hanc, Taylor and Tuleja, Am. J. Phys., 72(4), 2004 Symmetry Conservation Principle PCD-08
- 21. q1 q2 X v2 v1 Two positive charges q1 and q2 are moving along orthogonal directions as shown. They exert the Lorentz force q(E + vxB) on each other. The coulomb repulsion between them is directed away from each other, in opposite directions. The magnetic vxB force that the magnetic fields generated by the moving charges is however not in opposite directions. ACTION IS NOT OPPOSITE TO REACTION ! F12 ≠ - F21PCD-08
- 22. ( ) 1 2 12 21 1 2 0 ' d p d p dt dt p p Newton s III Law as statement of conservation of linear momentum = − = − + = uur uur r r ur ur F F d dt HOWEVER, WE HAVE JUST SEEN THAT ACTION IS NOT ALWAYS OPPOSITE TO REACTION ! F12 ≠ - F21 We shall see now that it is firmly placed on HOMOGENIETY of SPACE, thus expressing the relation between ‘SYMMETRY’ and ‘CONSERVATION LAWS’ (Noether’s theorm). Conservation of Momentum must be placed on a more robust principle. PCD-08
- 23. 1 N k k j j F f = =∑ ur uuur In an N-particle closed system, force on the kth particle is the sum of forces due to all other particles. WE SHALL NOT ASSUME WHETHER OR NOT F12 = (OR ≠) - F21 Consider ‘virtual’ displacement of the entire N-particle system in homogenous space. In such a displacement of the entire system in homogeneous space, the internal forces can do no work. PCD-08
- 24. 1 1 1 0 . . . . 0 N N N N k k k j k k j k dp dP s f s s s dt dP δ δ δ δ = = = = = = = ∑ ∑∑ ∑ uur ur uuur uuur uuur uuur uuur uuur ur F = dt dt Consider ‘virtual’ displacement of the entire N-particle system in homogenous space. Conservation of LINEAR MOMENTUM arises from HOMEGENEITY of SPACE. SYMMETRY CONSERVATION LAW Noether’s TheoremPCD-08
- 25. U1L3: Applications of Lagrange’s/Hamilton’s Equations Entire domain of Classical Mechanics Enables emergence of ‘Conservation of Energy’ and ‘Conservation of Momentum’ on the basis of a single principle. Symmetry Conservation Laws Governing principle: Variational principle – Principle of Least Action These methods have a charm of their own and very many applications…. Constraints / Degrees of Freedom - offers great convenience! ‘Action’ : dimensions ‘angular momentum’ : : :h Max Planck fundamental quantity in Quantum Mechanics We shall now illustrate the use of Lagrange’s / Hamilton’s equations to solve simple problems in Mechanics PCD-08
- 26. Manifestation of simple phenomena in different unrelated situations Dynamics of spring–mass systems, pendulum, oscillatory electromagnetic circuits, bio rhythms, share market fluctuations … radiation oscillators, molecular vibrations, atomic, molecular, solid state and nuclear physics, electrical engineering, mechanical engineering … Musical instruments ECG PCD-08
- 27. SMALL OSCILLATIONS 1581: Observations on the swaying chandeliers at the Pisa cathedral. Galileo (then only 17) recognized the constancy of the periodic time for small oscillations. PCD-08
- 28. : : : q q p & ( , , )L L q q t= & ( , , )H H q p t= Generalized Coordinate Generalized Velocity Generalized Momentum L p q ⎛ ⎞∂ = ⎜ ⎟ ∂⎝ ⎠& Use of Lagrange’s / Hamilton’s equations to solve the problem of Simple Harmonic Oscillator. PCD-08
- 29. ( , , ) 0 ' L L q q t Lagrangian L d L Lagrange s Equation q dt q = ⎛ ⎞∂ ∂ − =⎜ ⎟ ∂ ∂⎝ ⎠ & & ( , , ) ' k k k k k k k H q p L H H q p t Hamiltonian H H q p p q Hamilton s Equations = − = ∂ ∂ ∀ = = − ∂ ∂ ∑ & & &k: and 2nd order differential equation TWO 1st order differential equations PCD-08
- 30. 2 2 ( , , ) 2 2 0 ' 2 2 0 2 2 0 L L q q t Lagrangian m k L T V q q L d L Lagrange s Equation q dt q k d m q q dt kq mq mq kq Equation of Motion for a simple harmonic = = − = − ⎛ ⎞∂ ∂ − =⎜ ⎟ ∂ ∂⎝ ⎠ ⎛ ⎞ − − =⎜ ⎟ ⎝ ⎠ − − = = − & & & & && && oscillator 2nd order differential equation i.e. PCD-08
- 31. 2 2 2 2 2 2 2 2 2 ( , , ) ( , , ) : 2 2 2 2 2 2 2 2 ' L L q q t Lagrangian H H q p t Hamiltonian m k Lagrangian L T V q q L p mq q m k H pq L mq q q m k H q q p k H q m Hamilton s Equat = = = − = − ⎛ ⎞∂ = =⎜ ⎟ ∂⎝ ⎠ = − = − + = + = + & & & & & & & & ions of Motion for a simple harmonic oscillator 2 2 2 2 ( . . ) ' k H p p q p m m H k p q q i e f kq Hamilton s Equations TWO first order equations ∂ = = = ∂ ∂ = − = − ∂ = − & &and ( , , ) ( , , ) ! L L q q t H H q p t VERY IMPORTANT = = & p kq= −& 1 2 PCD-08
- 32. 2 2 2 2 : 2 2 2 2 m k Lagrangian L T V q q L p mq q p k H q m = − = − ⎛ ⎞∂ = =⎜ ⎟ ∂⎝ ⎠ = + & & & ( , , ) ( , , ) ! L L q q t H H q p t VERY IMPORTANT = = & Generalized Momentum is interpreted only as and not a product of mass with velocity L p q ⎛ ⎞∂ = ⎜ ⎟ ∂⎝ ⎠& Be careful about how you write the Lagrangian and the Hamiltonian for the Harmonic oscillator! PCD-08
- 33. In an INERTIAL frame of reference recognized first as one in which motion is self-sustaining, determined entirely by initial conditions alone, equilibrium denotes the state of rest or of uniform motion of an object along a straight line; motion at a constant angular momentum. Stable unstable neutral Absolute maximum Local vs. Absolute (Global) Extrema Local maximum Local minimum Absolute minimum Local minimum a bc e d in one dimension dU F dx = − PCD-08
- 34. Kinds of equilibrium unstable stable a bc e d stable unstable stable neutral neutral PCD-08
- 35. Is a point mid-way between two equal positive point charges a point of equilibrium for a unit point positive test charge? Can it be unambiguously classified as a point or ‘stable’ / ‘unstable’ equilibrium? what if the test charge is negative? + + Saddle point PCD-08
- 36. equilibria the point of inflexion the tangent cuts the curve neutral equilibrium is not possible for regular potential curves in one dimension direction of restoring force PCD-08
- 37. ( )2222 RyxKf −+= points of equilibria? equilibria Mexican hat www.CartoonStock.com Smoking is injurious to health and wealth consider points on the circle f = 0 push tangentially, push radially PCD-08
- 38. 0 0 0 2 3 2 3 0 0 0 02 3 1 1 ( ) ( ) ( ) ( ) + ( ) ... 2! 3!x x x U U U U x U x x x x x x x x x x ∂ ∂ ∂ = + − + − − + ∂ ∂ ∂ meaning of ‘small oscillations’ approximations 0 2 2 2 0 02 0 1 1 ( ) ( )+ ( ) = 2! 2 choosin ( ) 0 x U U x U x x x kx x by g U x ∂ ≈ − ∂ = dU F kx dx = − = − k x x m = −&& Potential for a Linear harmonic oscillator x U(x) The constant term is of no physical significance. It only adds a constant value to the potential and does not contribute to the physical force. PCD-08
- 39. dU F kx dx = − = − mx kx k x x m = − = − && && Potential for a Linear harmonic oscillator x U(x) PCD-08
- 40. l : length E: equilibrium S: support θ mg cosθl cos (1 cos ) h θ θ = − = − l l l 2 2 2 2 2 2 2 ( , , , ) 1 ( ) (1 cos ) 2 1 (1 cos ) 2 1 cos 2 L L r r L T V L m r r mg L m mg L m mg mg θ θ θ θ θ θ θ θ = = − = + − − = − − = − + && && l &l l &l l l (1 cos )V mgh mg θ= = −l Remember this! ALWAYS, the first thing to do is to set-up the Lagrangian in terms of the generalized coordinates and the generalized velocities. PCD-08
- 41. 2 21 cos 2 L m mg mgθ θ= − +&l l l 2 =0r L p r L p mlθ θ θ ∂ = ∂ ∂ = = ∂ & & & Subsequently, we can find the generalized momentum for each degree of freedom. : fixed lengthr = l 0 L d L q dt q ⎛ ⎞∂ ∂ − =⎜ ⎟ ∂ ∂⎝ ⎠& PCD-08
- 42. l : length E: equilibrium S: support θ mg cosθl cos (1 cos ) h θ θ = − = − l l l 2 2 ( , , , ) 1 cos 2 L L r r T V L m mg mg θ θ θ θ = = − = − + && &l l l 0. sin L r L mgl mglθ θ θ ∂ = ∂ ∂ = − ≈ − ∂ 2 =0r L p r L p mlθ θ θ ∂ = ∂ ∂ = = ∂ & & & 0 L d L q dt q ⎛ ⎞∂ ∂ − =⎜ ⎟ ∂ ∂⎝ ⎠& 2 2 ( ) 0 d mgl ml dt ml mgl g l θ θ θ θ θ θ − − = = − = − & && && 0 0 0 (1) (2) Solution: Substitute (2) in (1) i t i t q q q Ae Beω ω α ω α − = − = + ⇒ = && 0 g l ω = PCD-08
- 43. - -i i i i Hamiltonian Approach L H q L q p L q ⎡ ⎤∂ ⎡ ⎤= =⎢ ⎥ ⎣ ⎦∂⎣ ⎦ ∑ ∑& & & 2 21 cos 2 (cos ) (cos ) (cos ) ( sin ) H p m mg mg H mgl mgl mgl H mgl θθ θ θ θ θ θ θ θ θ θ θ θ = − + − ⎡ ⎤∂ ∂ ∂ = − ⎢ ⎥∂ ∂ ∂⎣ ⎦ = − − ≈ + ∂ ≈ ∂ & &l l l 2 p mlθ θ= & 2 p mlθ θ= &&& 2 H p ml mgl g l θ θ θ θ θ θ ∂ = = − = − ∂ = − &&& && H pθ θ ∂ = − ∂ & 0 g l ω = PCD-08
- 44. g l θ θ= −&& 0 g l ω = For the simple pendulum oscillating in the gravitational field where the acceleration due to gravity is g, we must, and do, get the same answer regardless of which approach we employ: (1) Newtonian (2) Lagrangian (3) Hamiltonian Note! We haven’t used ‘force’, ‘tension in the string’ etc. in the Lagrangian and Hamiltonian approach! PCD-08
- 45. Uniform circular motion and SHM www.physics.uoguelph.ca www.answers.com 2ω πν= Intrinsic natural frequency ‘reference circle’ for the Simple Harmonic Oscillator.PCD-08
- 46. Qmax Qmin I=0 I Imax I I=0 PCD-08
- 47. max 2 2 ( ) is proportional to , not to V, as in the case of a resistor. mac L Q V C dI d Q V L L LQ dt dt d d dV I Q Q CV C dt dt dt dV I dt = = − = − = − = = = = && & Unlike what happens in a resistor the current and voltage in an inductance L and in a capacitor C do not peak together. CV LV I Voltage lags the current in a capacitor by 900, but leads the current in an inductor by the same amount. 2 2 0 0 1 ( ) L CV V d Q Q L dt C Q Q LC − + = + + = = −&& PCD-08
- 48. 0 0 0 1 ( ) (1) (2) Most general solution: Substitute (2) in (1) i t i t k x x m Q Q LC q q q Ae Beω ω α ω α − = − = − = − = + ⇒ = && && && 0 0 1 k m LC ω ω = = Electro-mechanical analogues: Inductance mass, inertia Capacitance 1/k, compliance Question: Could we have associated L with 1/k and C with m? PCD-08
- 49. Linear relation between restoring force and displacement for spring-mass system: Hooke’s law, after Robert Hooke (1635-1703), (a contemporary of Newton), who empirically discovered this relation for several elastic materials in 1678. k x x m = −&& http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Hooke.html PCD-08