![xiv Preface
pend on context, and often even change within a given context.1
For example, a fundamental result of mechanics is the Lagrange
equations. Using traditional notation the Lagrange equations are
written
d
dt
∂L
∂q̇i
−
∂L
∂qi
= 0.
The Lagrangian L must be interpreted as a function of the position
and velocity components qi and q̇i, so that the partial deriva-
tives make sense, but then in order for the time derivative d/dt
to make sense solution paths must have been inserted into the
partial derivatives of the Lagrangian to make functions of time.
The traditional use of ambiguous notation is convenient in simple
situations, but in more complicated situations it can be a serious
handicap to clear reasoning. In order that the reasoning be clear
and unambiguous, we have adopted a more precise mathematical
notation. Our notation is functional and follows that of modern
mathematical presentations.2
Computation also enters into the presentation of the mathe-
matical ideas underlying mechanics. We require that our mathe-
matical notations be explicit and precise enough so that they can
1
In his book on mathematical pedagogy [15], Hans Freudenthal argues that
the reliance on ambiguous, unstated notational conventions in such expressions
as f(x) and df(x)/dx makes mathematics, and especially introductory calcu-
lus, extremely confusing for beginning students; and he enjoins mathematics
educators to use more formal modern notation.
2
In his beautiful book Calculus on Manifolds (1965), Michael Spivak uses
functional notation. On p.44 he discusses some of the problems with classical
notation. We excerpt a particularly juicy quote:
The mere statement of [the chain rule] in classical notation requires the
introduction of irrelevant letters. The usual evaluation for D1(f ◦(g, h))
runs as follows:
If f(u, v) is a function and u = g(x, y) and v = h(x, y) then
∂f(g(x, y), h(x, y))
∂x
=
∂f(u, v)
∂u
∂u
∂x
+
∂f(u, v)
∂v
∂v
∂x
[The symbol ∂u/∂x means ∂/∂x g(x, y), and ∂/∂u f(u, v) means
D1f(u, v) = D1f(g(x, y), h(x, y)).] This equation is often written simply
∂f
∂x
=
∂f
∂u
∂u
∂x
+
∂f
∂v
∂v
∂x
.
Note that f means something different on the two sides of the equation!](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-18-320.jpg)
![Preface xv
be interpreted automatically, as by a computer. As a consequence
of this requirement the formulas and equations that appear in the
text stand on their own. They have clear meaning, independent of
the informal context. For example, we write Lagrange’s equations
in functional notation as follows:3
D(∂2L ◦ Γ[q]) − ∂1L ◦ Γ[q] = 0
The Lagrangian L is a real-valued function of time t, coordinates
x, and velocities v; the value is L(t, x, v). Partial derivatives
are indicated as derivatives of functions with respect to partic-
ular argument positions; ∂2L indicates the function obtained by
taking the partial derivative of the Lagrangian function L with
respect to the velocity argument position. The traditional partial
derivative notation, which employs a derivative with respect to a
“variable,” depends on context and can lead to ambiguity.4 The
partial derivatives of the Lagrangian are then explicitly evaluated
along a path function q. The time derivative is taken and the
Lagrange equations formed. Each step is explicit; there are no
implicit substitutions.
Computational algorithms are used to communicate precisely
some of the methods used in the analysis of dynamical phenomena.
Expressing the methods of variational mechanics in a computer
language forces them to be unambiguous and computationally
effective. Computation requires us to be precise about the repre-
sentation of mechanical and geometric notions as computational
objects and permits us to represent explicitly the algorithms for
manipulating these objects. Also, once formalized as a procedure,
a mathematical idea becomes a tool that can be used directly to
compute results.
Active exploration on the part of the student is an essential
part of the learning experience. Our focus is on understanding
the motion of systems; to learn about motion the student must
actively explore the motion of systems through simulation and
3
This is presented here without explanation, to give the flavor of the notation.
The text gives a full explanation.
4
“It is necessary to use the apparatus of partial derivatives, in which even the
notation is ambiguous.” From V.I. Arnold, Mathematical Methods of Classical
Mechanics (1980), Section 47, p258. See also the footnote on that page.](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-19-320.jpg)
 =
Z t2
t1
F[γ] (1.1)
where F[γ] is a function of time that measures some local property
of the path. It may depend upon the value of the function γ at
that time and the value of any derivatives of γ at that time.8
The configuration path can be locally described at a moment in
terms of the configuration, the rate of change of the configuration,
and all the higher derivatives of the configuration at the given
moment. Given this information the path can be reconstructed in
some interval containing that moment.9 Local properties of paths
can depend on no more than the local description of the path.
The function F measures some local property of the configura-
tion path γ. We can decompose F[γ] into two parts: a part that
measures some property of a local description and a part that ex-
tracts a local description of the path from the path function. The
function that measures the local property of the system depends
on the particular physical system; the method of construction of a
local description of a path from a path is the same for any system.
We can write F[γ] as a composition of these two functions:10
F[γ] = L ◦ T [γ]. (1.2)
7
A definite integral of a real-valued function f of a real argument is written
R b
a
f. This can also be written
R b
a
f(x)dx. The first notation emphasizes that
a function is being integrated.
8
Traditionally, square brackets are put around functional arguments. In this
case, the square brackets remind us that the value of S may depend on the
function γ in complicated ways, such as through its derivatives.
9
In the case of a real-valued function the value of the function and its deriva-
tives at some point can be used to construct a power series. For sufficiently
nice functions (real analytic) the power series constructed in this way con-
verges in some interval containing the point. Not all functions can be locally
represented in this way. For example, the function f(x) = exp(−1/x2
), with
f(0) = 0, is zero and has all derivatives zero at x = 0, but this infinite number
of derivatives is insufficient to determine the function value at any other point.
10
Here ◦ denotes composition of functions: (f◦g)(t) = f(g(t)). In our notation
the application of a path-dependent function to its path is of higher precedence
than the composition, so L ◦ T [γ] = L ◦ (T [γ]).](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-32-320.jpg)
 = (t, γ(t), Dγ(t), . . .) (1.3)
We refer to this tuple, which includes as many derivatives as are
needed, as the local tuple.
The function L depends on the specific details of the physical
system being investigated, but does not depend on any particular
configuration path. The function L computes a real-valued local
property of the path. We will find that L needs only a finite num-
ber of components of the local tuple to compute this property:
The path can be locally reconstructed from the full local descrip-
tion; that L depends on a finite number of components of the local
tuple guarantees that it measures a local property.12
The advantage of this decomposition is that the local descrip-
tion of the path is computed by a uniform process from the con-
figuration path, independent of the system being considered. All
of the system-specific information is captured in the function L.
The function L is called a Lagrangian13 for the system, and the
resulting action,
S[γ](t1, t2) =
Z t2
t1
L ◦ T [γ], (1.4)
11
The derivative Dγ of a configuration path γ can be defined in terms of
ordinary derivatives by specifying how it acts on sufficiently smooth real-
valued functions f of configurations. The exact definition is unimportant at
this stage. If you are curious see footnote 23.
12
We will later discover that an initial segment of the local tuple will be
sufficient to determine the future evolution of the system. That a configuration
and a finite number of derivatives determines the future means that there is
a way of determining all of the rest of the derivatives of the path from the
initial segment.
13
The classical Lagrangian plays a fundamental role in the path-integral for-
mulation of quantum mechanics (due to Dirac and Feynman), where the com-
plex exponential of the classical action yields the relative probability ampli-
tude for a path. The Lagrangian is the starting point for the Hamiltonian
formulation of mechanics (discussed in chapter 3), which is also essential in
the Schrödinger and Heisenberg formulations of quantum mechanics and in
the Boltzmann-Gibbs approach to statistical mechanics.](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-33-320.jpg)
 is stationary with respect to variations of the path.
For Lagrangians that depend only on the configuration and rate
of change of configuration the variations are restricted to those
that preserve the configurations at t1 and t2.15
14
The principle is often called the “Principle of Least Action” because its
initial formulations spoke in terms of the action being minimized rather than
the more general case of taking on a stationary value. The term “Principle of
Least Action” is also commonly used to refer to a result, due to Maupertuis,
Euler, and Lagrange, which says that free particles move along paths for which
the integral of the kinetic energy is minimized among all paths with the given
endpoints. Correspondingly, the term “action” is sometimes used to refer
specifically to the integral of the kinetic energy. (Actually, Euler and Lagrange
used the vis viva, or twice the kinetic energy.)
15
Other ways of stating the principle of stationary action make it sound teleo-
logical and mysterious. For instance, one could imagine that the system con-
siders all possible paths from its initial configuration to its final configuration
and then chooses the one with the smallest action. Indeed, the underlying vi-
sion of a purposeful, economical, and rational universe played no small part in
the philosophical considerations that accompanied the initial development of](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-34-320.jpg)
![1.2 Configuration Spaces 9
Exercise 1.1: Fermat optics
Fermat observed that the laws of reflection and refraction could be ac-
counted for by the following facts: Light travels in a straight line in any
particular medium with a velocity that depends upon the medium. The
path taken by a ray from a source to a destination through any sequence
of media is a path of least total time, compared to neighboring paths.
Show that these facts do imply the laws of reflection and refraction.16
1.2 Configuration Spaces
Let us consider mechanical systems that can be thought of as
composed of constituent point particles, with mass and position,
but with no internal structure.17 Extended bodies may be thought
of as composed of a large number of these constituent particles
with specific spatial relationships between them. Extended bodies
maintain their shape because of spatial constraints between the
constituent particles. Specifying the position of all the constituent
particles of a system specifies the configuration of the system. The
existence of constraints between parts of the system, such as those
that determine the shape of an extended body, means that the
constituent particles cannot assume all possible positions. The
set of all configurations of the system that can be assumed is
called the configuration space of the system. The dimension of the
mechanics. The earliest action principle that remains part of modern physics is
Fermat’s Principle, which states that the path traveled by a light ray between
two points is the path that takes the least amount of time. Fermat formu-
lated this principle around 1660 and used it to derive the laws of reflection
and refraction. Motivated by this, the French mathematician and astronomer
Pierre-Louis Moreau de Maupertuis enunciated the Principle of Least Action
as a grand unifying principle in physics. In his Essai de cosmologie (1750)
Maupertuis appealed to this principle of “economy in nature” as evidence of
the existence of God, asserting that it demonstrated “God’s intention to regu-
late physical phenomena by a general principle of the highest perfection.” For
a historical perspective of Maupertuis’s, Euler’s, and Lagrange’s roles in the
formulation of the principle of least action, see Jourdain [25].
16
For reflection the angle of incidence is equal to the angle of reflection. Re-
fraction is described by Snell’s law. Snell’s Law is that when light passes from
one medium to another, the ratio of the sines of the angles made to the normal
to the interface is the inverse of the ratio of the refractive indices of the media.
The refractive index is the ratio of the speed of light in the vacuum to the
speed of light in the medium.
17
We often refer to a point particle with mass but no internal structure as a
point mass.](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-35-320.jpg)
![1.3 Generalized Coordinates 13
the values χi(m) of the coordinate functions are the generalized
coordinates of the configuration. These generalized coordinates
permit us to identify points of the n-dimensional configuration
space with n-tuples of real numbers.21 For any given configura-
tion space, there are a great variety of ways to choose generalized
coordinates. Even for a single point moving without constraints,
we can choose rectangular coordinates, polar coordinates, or any
other coordinate system that strikes our fancy.
The motion of the system can be described by a configuration
path γ mapping time to configuration-space points. Correspond-
ing to the configuration path is a coordinate path q = χ◦γ mapping
time to tuples of generalized coordinates. If there is more than
one degree of freedom the coordinate path is a structured object:
q is a tuple of component coordinate path functions qi = χi ◦ γ.
At each instant of time t, the values q(t) = (q0(t), . . . , qn−1(t)) are
the generalized coordinates of a configuration.
The derivative Dq of the coordinate path q is a function22 that
gives the rate of change of the configuration coordinates at a given
time: Dq(t) = (Dq0(t), . . . , Dqn−1(t)). The rate of change of a
generalized coordinate is called a generalized velocity.
We can make coordinate representations for higher derivatives
of the path as well. We introduce the function (pronounced
21
More precisely, the generalized coordinates identify open subsets of the con-
figuration space with open subsets of Rn
. It may require more than one set of
generalized coordinates to cover the entire configuration space. For example,
if the configuration space is a two-dimensional sphere, we could have one set
of coordinates that maps (a little more than) the northern hemisphere to a
disk, and another set that maps (a little more than) the southern hemisphere
to a disk, with a strip near the equator common to both coordinate systems.
A space that can be locally parametrized by smooth coordinate functions is
called a differentiable manifold. The theory of differentiable manifolds can be
used to formulate a coordinate-free treatment of variational mechanics. An
introduction to mechanics from this perspective can be found in [2] or [5] .
22
The derivative of a function f is a function. It is denoted Df. Our notational
convention is that D is a high-precedence operator. Thus D operates on the
adjacent function before any other application occurs: Df(x) is the same as
(Df)(x).](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-39-320.jpg)
 = (t, q(t), Dq(t), . . .) . (1.6)
The evaluation of Γ only involves taking derivatives of the coordi-
nate path q = χ ◦ γ; the function Γ does not depend on χ. From
relations (1.5) and (1.6) we find
Γ[q] = χ ◦ T [γ]. (1.7)
Exercise 1.3: Generalized coordinates
For each of the systems described in exercise 1.2 specify a system of
generalized coordinates that can be used to describe the behavior of the
system.
Lagrangians in generalized coordinates
The action is a property of a configuration path segment for a
particular Lagrangian L. The action does not depend on the co-
ordinate system that is used to label the configurations. We can
use this property to find a coordinate representation Lχ for the
Lagrangian L.
23
The formal definition of is unimportant to the discussion, but if you really
want to know here is one way to do it:
First, we define the derivative Dγ of a configuration path γ in terms of
ordinary derivatives by specifying how it acts on sufficiently smooth real-
valued functions f of configurations: (Dn
γ)(t)(f) = Dn
(f ◦ γ)(t). Then we
define χ(a, b, c, d, . . .) = (a, χ(b), c(χ), d(χ), . . .) . With this definition:
χ(t, γ(t), Dγ(t), D2
γ(t), . . .) =
¡
t, χ(γ(t)), Dγ(t)(χ), D2
γ(t)(χ), . . .
¢
=
¡
t, χ ◦ γ(t), D(χ ◦ γ)(t), D2
(χ ◦ γ)(t), . . .
¢
=
¡
t, q(t), Dq(t), D2
q(t), . . .
¢
.](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-40-320.jpg)
 =
Z t2
t1
L ◦ T [γ]. (1.8)
The Lagrangian L is a function of the local tuple T [γ](t) =
(t, γ(t), Dγ(t), . . .). The local tuple has the coordinate represen-
tation Γ[q] = χ ◦ T [γ], where q = χ ◦ γ. So if we choose24
Lχ = L ◦ −1
χ , (1.9)
then25
Lχ ◦ Γ[q] = L ◦ T [γ]. (1.10)
On the left we have the composition of functions that use the
intermediary of a coordinate representation; on the right we have
the composition of two functions that do not involve coordinates.
We define the coordinate representation of the action to be
Sχ[q](t1, t2) =
Z t2
t1
Lχ ◦ Γ[q]. (1.11)
The function Sχ takes a coordinate path; the function S takes a
configuration path. Since the integrands are the same by equa-
tion (1.10) the integrals have the same value:
S[γ](t1, t2) = Sχ[χ ◦ γ](t1, t2). (1.12)
So we have a way of constructing coordinate representations of a
Lagrangian that gives the same action for a path in any coordinate
system.
For Lagrangians that depend only on positions and velocities
the action can also be written
Sχ[q](t1, t2) =
Z t2
t1
Lχ (t, q(t), Dq(t)) dt. (1.13)
24
The coordinate function χ is locally invertible, and so is χ.
25
L ◦ T [γ] = L ◦
−1
χ ◦ χ ◦ T [γ] = Lχ ◦ Γ[χ ◦ γ] = Lχ ◦ Γ[q].](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-41-320.jpg)
.
Let q + ²η be a nearby path, obtained from q by adding a path
36
Surely for a real physical situation we would have to specify units for these
quantities. In this illustration we do not give units.
37
Here we use decimal numerals to specify the parameters. This forces the
representations to be floating point, which is efficient for numerical calculation.
If symbolic algebra is to be done it is essential that the numbers be exact
integers or rational fractions, so that expressions can be reliably reduced to
lowest terms. Such numbers are specified without a decimal point.
38
The squared magnitude of the velocity is ~
v · ~
v, the vector dot-product of
the velocity with itself. The square of a structure of components is defined to
be the sum of the squares of the individual components, so we write simply
v2
= v · v.](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-46-320.jpg)
. Euler and Lagrange found S[q +
²η](t1, t2) > S[q](t1, t2) for any η that is zero at the endpoints and
for any small non-zero ².
Let’s check this numerically by varying the test path, adding
some amount of a test function that is zero at the endpoints t = t1
and t = t2. To make a function η that is zero at the endpoints,
given a sufficiently well-behaved function ν, we can use η(t) =
(t − t1)(t − t2)ν(t). This can be implemented:
(define ((make-eta nu t1 t2) t)
(* (- t t1) (- t t2) (nu t)))
We can use this to compute the action for a free particle over a
path varied from the given path, as a function of ²:40
(define ((varied-free-particle-action mass q nu t1 t2) epsilon)
(let ((eta (make-eta nu t1 t2)))
(Lagrangian-action (L-free-particle mass)
(+ q (* epsilon eta))
t1
t2)))
The action for the varied path, with ν(t) = (sin t, cos t, t2), and
² = 0.001 is, as expected, larger than for the test path:
((varied-free-particle-action 3.0 test-path
(up sin cos square)
0.0 10.0)
0.001)
436.29121428571153
39
Note that we are doing arithmetic on functions. We extend the arithmetic
operations so that the combination of two functions of the same type (same
domains and ranges) is the function on the same domain that combines the
values of the argument functions in the range. For example, if f and g are
functions of t, then fg is the function t 7→ f(t)g(t). A constant multiple of
a function is the function whose value is the constant times the value of the
function for each argument: cf is the function t 7→ cf(t).
40
Note that we are adding procedures. Paralleling our extension of arithmetic
operations to functions, arithmetic operations are extended to compatible pro-
cedures.](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-47-320.jpg)
 is stationary (with respect to any
variation in the path that keeps the endpoints of the path fixed)
then
D(∂2L ◦ Γ[q]) − ∂1L ◦ Γ[q] = 0. (1.18)
Here L is a real-valued function of a local tuple; ∂1L and ∂2L
denote the partial derivatives of L with respect to its general-
ized position and generalized velocity arguments.49 The function
∂2L maps a local tuple to a structure whose components are the
derivatives of L with respect to each component of the gener-
alized velocity. The function Γ[q] maps time to the local tuple:
Γ[q](t) = (t, q(t), Dq(t), . . .). Thus the compositions ∂1L◦Γ[q] and
48
This result was initially discovered by Euler and later rederived by Lagrange.
49
The derivative or partial derivative of a function that takes structured argu-
ments is a new function that takes the same number and type of arguments.
The range of this new function is itself a structure with the same number of
components as the argument with respect to which the function is differenti-
ated.](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-52-320.jpg)
![1.5.1 Derivation of the Lagrange Equations 27
∂2L◦Γ[q] are functions of one argument, time. The Lagrange equa-
tions assert that the derivative of ∂2L ◦ Γ[q] is equal to ∂1L ◦ Γ[q],
at any time. Given a Lagrangian, the Lagrange equations form a
system of ordinary differential equations that must be satisfied by
realizable paths.50
1.5.1 Derivation of the Lagrange Equations
We will show that Principle of Stationary Action implies that
realizable paths satisfy a set of ordinary differential equations.
First we will develop tools for investigating how path-dependent
functions vary as the paths are varied. We will then apply these
tools to the action, to derive the Lagrange equations.
Varying a path
Suppose that we have a function f[q] that depends on a path q.
How does the function vary as the path is varied? Let q be a
coordinate path and q + ²η be a varied path, where the function
η is a path-like function that can be added to the path q, and the
factor ² is a scale factor. We define the variation δηf[q] of the
function f on the path q by51
δηf[q] = lim
²→0
µ
f[q + ²η] − f[q]
²
¶
. (1.19)
50
Lagrange’s equations are traditionally written in the form
d
dt
∂L
∂q̇
−
∂L
∂q
= 0,
or, if we write a separate equation for each component of q, as
d
dt
∂L
∂q̇i
−
∂L
∂qi
= 0 i = 0, . . . , n − 1 .
In this way of writing Lagrange’s equations the notation does not distinguish
between L, which is a real-valued function of three variables (t, q, q̇), and L ◦
Γ[q], which is a real-valued function of one real variable t. If we do not realize
this notational pun, the equations don’t make sense as written—∂L/∂q̇ is a
function of three variables, so we must regard the arguments q, q̇ as functions
of t before taking d/dt of the expression. Similarly, ∂L/∂q is a function of
three variables, which we must view as a function of t before setting it equal
to d/dt(∂L/∂q̇). These implicit applications of the chain rule pose no problem
in performing hand computations—once you understand what the equations
represent.
51
The variation operator δη is like the derivative operator in that it acts on
the immediately following function: δηf[q] = (δηf)[q].](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-53-320.jpg)
![28 Chapter 1 Lagrangian Mechanics
The variation of f is a linear approximation to the change in the
function f for small variations in the path. The variation of f
depends on η.
A simple example is the variation of the identity path function:
I[q] = q. Applying the definition
δηI[q] = lim
²→0
µ
(q + ²η) − q
²
¶
= η. (1.20)
It is traditional to write δηI[q] simply as δq. Another example is
the variation of the path function that returns the derivative of
the path. We have
δηg[q] = lim
²→0
µ
D(q + ²η) − Dq
²
¶
= Dη with g[q] = Dq. (1.21)
It is traditional to write δηg[q] as δDq.
The variation may be represented in terms of a derivative. Let
g(²) = f[q + ²η], then
δηf[q] = lim
²→0
µ
g(²) − g(0)
²
¶
= Dg(0). (1.22)
Variations have the following derivative-like properties. For
path-dependent functions f and g and constant c:
δη(f g)[q] = δηf[q] g[q] + f[q] δηg[q] (1.23)
δη(f + g)[q] = δηf[q] + δηg[q] (1.24)
δη(cf)[q] = c δηf[q]. (1.25)
Let F be a path-independent function and let g be a path-dependent
function, then
δηh[q] = (DF ◦ g[q]) δηg[q] with h[q] = F ◦ g[q]. (1.26)
The operators D (differentiation) and δ (variation) commute in
the following sense:
Dδηf[q] = δηg[q] with g[q] = D(f[q]). (1.27)
Variations also commute with integration in a similar sense.
If a path-dependent function f is stationary for a particular
path q with respect to small changes in that path then it must be](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-54-320.jpg)
![land enclosed bore to the whole area of the county, but the
proportion which it bore to the whole area available for cultivation.
This, which is of course not ascertainable, is clearly a very different
thing.[462] It is no consolation to a family which has been evicted
from a prosperous farm to be told that it can settle on a moor or a
marsh, on Blackstone Edge or Deeping Fen. To argue that enclosing
was of little consequence, because so small a proportion of the total
land area was enclosed, is almost precisely similar to arguing that
overcrowding is of little consequence, because the area of Great
Britain divided by the population gives a quotient of about one and a
half acres to every human being in the country. The evidence of a
general trend of opinion during a century and a half—opinion by no
means confined to the peasants, or to the peasants' champions like
Hales, or to idealists like Sir Thomas More, or to the preachers of
social righteousness like Latimer and Crowley, but shared by Wolsey
and Thomas Cromwell in the earlier part of the century, Robert Cecil
and Francis Bacon[463] at the end of it—to the effect that the
agrarian changes caused extensive depopulation, is really a firmer
basis for judging their effects than are statistics which, however
carefully worked up, are necessarily unreliable, and which, when
reliable, are not quite the statistics required. When that opinion is
backed by documentary proof that from one village thirty persons,
from another fifty, from another the whole population, were
displaced, though of course we cannot say that such displacement
was general, we can say that it was not unknown, and that if
contemporaries were guilty of exaggeration (as they probably were),
their exaggeration took the form not of inventing extreme cases, but
of suggesting that such extreme cases were the rule. On the whole,
therefore, our conclusions as to the quantitative measurement of
depopulation caused in the sixteenth century must still, in spite of
the researches of Mr. Leadam and Professor Gay, be a negative one.
In the first place, we cannot say, even approximately, what
proportion of the total landholding population was displaced. In the
second place, such figures as we do possess are not of a kind to
outweigh the direct evidence of contemporary observers that the](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-56-320.jpg)
![things possessing many 'plausible advantages.' Legislation for the
management of the Poor often impeded, and only occasionally
expedited, this beneficent process.... It proceeded from ignorance of
the true nature of progress, and from a denial or neglect of the
power of absorption possessed by a free society.”[464] It is obvious
that in this passage Mr. Mackay uses his interpretation of Poor Law
origins to make a very trenchant criticism upon the whole principle
involved in the public maintenance of the destitute. That principle
was not introduced because new conditions made its adoption
indispensable. It survived from an older order of things into a world
in which the only serious causes of destitution are personal and not
economic, and in which therefore it is quite inappropriate. To tolerate
it is to drag for ever a clanking chain, one end of which is fastened
round the bleeding ankles of modern society, and the other anchored
in the hideous provisions of the Statute of Labourers. Nor should we
be wrong if we said that a similar theory, though less lucidly
expressed, has had a considerable influence upon Poor Law practice.
For the idea of a Poor Law as an anachronism which is quite out of
place in a developed economic society is implied more than once in
the celebrated report drafted by Senior and Chadwick in 1834, and
has passed from that brilliant piece of special pleading into the minds
of three generations of administrators. “A person,” they state, “who
attributes pauperism to the inability to procure employment, will
doubt the efficiency of the cause which we propose to remove it,"
whereas “whenever inquiries have been made as to the previous
condition of the able-bodied individuals who live in such numbers on
the town parishes, it has been found that the pauperism of the
greater number has originated in indolence, improvidence, and vice,
and might have been avoided by ordinary care and industry. The
majority of the Statutes connected with the administration of public
relief have created new evils, and aggravated those which they were
intended to prevent.”[465]
A discussion of Poor Law theory and history falls outside the limits of
this essay. But in forming an estimate of the effects of the agrarian
changes which have been described above, it is perhaps not out of](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-58-320.jpg)
![scourged and buffeted by all men. The destitution of the aged and
impotent, of fatherless children and widows, is familiar enough. It
has been with the world from time immemorial. It has been for
centuries the object of voluntary charitable effort; and when the
dissolution of the monasteries dries up one great channel of
provision, the Government intervenes with special arrangements[466]
to take their place a whole generation before it can be brought to
admit that there is any problem of the unemployed, other than the
problem of the sturdy rogue. The distinction between the able-
bodied unemployed and the impotent is one which is visible to the
eye of sense. The distinction between the man who is unemployed
because he cannot get work and the man who is unemployed
because he does not want work, requires a modicum of knowledge
and reflection which even at the present day is not always
forthcoming. The former distinction, therefore, is not supplemented
by the latter until the beginning of the last quarter of the century.
[467] In one respect, that of the Law of Settlement, the English Poor
Law does show traces of a mediæval origin. In all other respects, so
far from being a survival from the Middle Ages, it comes into
existence just at the time when mediæval economic conditions are
disappearing. It is not accepted at once as a matter of course that
the destitute shall be publicly relieved, still less that the able-bodied
destitute deserve anything but punishment. Governments make
desperate efforts for about one hundred years to evade their new
obligations. They whip and brand and bore ears; they offer the
vagrant as a slave to the man who seizes him; they appeal to
charity; they introduce the parish clergy to put pressure on the
uncharitable; they direct the bishops to reason with those who stop
their ears against the parish clergy. When merely repressive
measures and voluntary effort are finally discredited, they levy a
compulsory charge rather as a fine for contumacy than as a rate,
and slide reluctantly into obligatory assessments[468] only when all
else has failed. And if we ask why the obligation of maintaining the
destitute should have received national recognition first in the
sixteenth century, we can only answer by pointing to that trend away
from the stationary conditions of agriculture to the fluctuating](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-60-320.jpg)
![conditions of trade, and in particular to that displacement of the rural
population, which we have already seen was one result of enclosure.
The national Poor Law is not a mediæval anachronism. It is the
outcome of conditions which seem to the men of the sixteenth
century new and appalling. Of these conditions the most important
are the agrarian changes.
Let us try for a moment to put ourselves in the position of a family
which has been evicted from its holding to make room for sheep.
When the last stick of furniture has been tumbled out by the bailiff,
where, poor houseless wretches, are they to turn? They cannot get
work in their old home, even if they can get lodgings, for the
attraction of sheep-farming is that the wage bill is so low. Will they
emigrate from England like the Scotch crofters? There are people
who in the seventeenth century will advise them to seek a haven
with the godly folk who have crossed the Atlantic, who will argue
that England is overstocked, that “there is such pressing and
oppressing in town and country about farms, trades, traffic, so as a
man can hardly anywhere set up a trade but he shall pull down two
of his neighbours,” and point out that “the country is replenished
with new farmers, and the almhouses are filled with old labourers,”
that “the rent-taker lives on sweet morsels, but the rent-payer eats a
dry crust often with watery eyes.”[469] But enclosures have been
going on for a century before the plantations exist to offer a refuge,
and in any case the probability of the country folk hearing of them is
very remote. Can a man migrate to seek work in another part of the
country? Not easily, for, apart from the enormous practical
difficulties, the law puts obstacles in his way, and the law is backed
up with enthusiasm by every parish and town in the country. There
are three possible attitudes which a State may adopt towards the
questions arising from the ebb and flow of population. It may argue,
with the optimists of 1834, that the mobility of labour is a good
thing, a symptom of alertness and energy, and that it will take place
of itself to the extent which is economically desirable, provided that
no impediments are placed in the way of those who desire to better
themselves by looking for work elsewhere. Or, while believing that it](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-61-320.jpg)
![is much to be desired that people should migrate freely from place to
place in search of employment, it may nevertheless reflect that the
mere absence of restrictions does not in fact stimulate such
movement, and therefore take upon itself its encouragement through
the publication of information and the registration of unemployed
workers. Or, subordinating economic to political considerations, it
may hold that the movement of a large number of unemployed
persons up and down the country is not an indication of a
praiseworthy spirit of enterprise, but a menace to public order which
must be sternly repressed. We need hardly say that this last view is
the one characteristic of the sixteenth century. The attitude towards
the man on tramp in search of employment is exactly the opposite of
that which is held at the present day. He is not less, but much more,
culpable than he who remains in his own parish and lives on his
neighbours. He is assumed not to be seeking work but to be
avoiding it, and avoiding it in a restless and disorderly manner. Hear
what the worthy Harrison says when the State has already made the
provision for the unemployed a charge upon each parish:—“But if
they refuse to be supported by this benefit of the law, and will rather
endeavour by going to and fro to maintain their idle trades, then are
they adjudged to be parcel of the third sort (i.e. wilful vagrants), and
so, instead of courteous refreshing at home, are often corrected with
sharp execution and whip of justice abroad. Many there are which,
notwithstanding the rigour of the laws provided on that behalf, yield
rather with this liberty (as they call it) to be daily under the fear and
terror of the whip, than by abiding where they were born or bred, to
be provided for by the devotion of the parishes.”[470] The village is
still thought of as the unit of employment. It is still regarded as
being equipped with the means of finding work for all its inhabitants,
as though there had been no movement towards pasture-farming to
prick a hole in its economic self-sufficiency. The presumption,
therefore, is against the man who leaves the parish where he is
known to his neighbours. He must prove that he is going to take up
work for which he is already engaged. He must get a licence from his
last employer. As far as the able-bodied are concerned the Poor Law
is in origin a measure of social police. Relief is thrown in as a](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-62-320.jpg)
![makeweight, because by the end of the sixteenth century our
statesmen have discovered that when economic pressure reaches a
certain point they cannot control men without it. The whip has no
terrors for the man who must look for work or starve. So every
Sunday after church, while Parson’s sermon is still fresh in our
minds, we board out our poor by rotation “among such householders
as will maintain them meat and work and such wages as they shall
deserve for the week following.”[471] Heaven help us if the next
parish does not do the same!
And the Poor Law is a police measure for the necessity of which the
agrarian changes are largely responsible. In spite of all the obstacles
in the way of migration, in spite of whip and courteous refreshment,
men do in fact migrate, and not only men, but women and children.
By the latter part of the century, at any rate, statesmen have begun
to understand that pauperism and vagrancy stand to the
depopulation caused by enclosure in the relation of effect to cause.
The revolution in the official attitude to the problem caused by this
belated illumination is as great as that which has taken place in the
last ten years with regard to unemployment. Once the new
standpoint has been seized, though opinion, and the opinion not only
of the ruling classes, but of burgesses and villagers, still treats the
vagrant with iron severity, it never quite relapses into the
comfortable doctrine, the grand discovery of a commercial age, that
distress is itself a proof of the demerits of its victim, and that
Heaven, like a Utilitarian philosopher, permits the existence of
destitution only that it may make “less eligible" the lot of
“improvidence and vice.” It is saved from this last error not by the
lore of economists, but because it regards economic questions
through the eyes of a sturdy matter-of-fact morality. It is sufficiently
enlightened to recognise that even among vagrants there is a class
which is more sinned against than sinning, a class of whom it can be
asked “at whose hands shall the blood of these men be
required?”[472] It is sufficiently ingenuous to answer by pointing to
“some covetous man" who, “espying a further commodity in their
commons, holds, and tenures, doth find such means as thereby to](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-63-320.jpg)
![wipe many out of their occupyings and turn the same unto his
private gains.”[473] Occasionally the effect of enclosures is brought
home to the encloser in a practical way, by compelling him not only
to pay a fine to the Crown, but also to make a contribution towards
the relief of the poor whose numbers he has increased.[474]
To see the way in which the relation between the problems of
pauperism and of agrarian depopulation is regarded, turn to the
debates in the House of Commons. In the year 1597, when both
questions are acute (the preceding year had seen a recrudescence of
agrarian rioting), a member or minister, probably Robert Cecil, is
preparing notes for a speech[475] on the subject in Parliament. What
are the points he emphasises? They are the high price of corn
caused by bad harvests and the manipulations of middlemen, the
enclosing of land and the conversion of arable to pasture, which
naturally intensifies the difficulty of securing adequate food supplies,
“the decaying and plucking down of houses, ... and not only the
plucking down of some few houses, but the depopulating of whole
towns ... and keeping of a shepherd only, whereby many subjects
are turned without habitation, and fill the country with rogues and
idle persons.” When Parliament meets in October, the House is at
once busy with different aspects of the same question.[476] Bills are
introduced dealing with forestallers, regrators, and engrossers of
corn, with vagrancy and pauperism, and with enclosures, and a
committee is appointed to consider the latter question. In the
debates which follow there is the usual division of opinion between
the champions of economic reform and the advocates of more, and
more ruthless, “deterrence," between those who wish to legislate as
to causes and those who are mainly occupied with symptoms.
Bacon, master as ever of the science of his subject, insists with
invincible logic that pauperism is one part of the general agrarian
problem, and he is supported by Robert Cecil. On the other hand,
the experts as to pauperism—we can imagine the county justices
fresh from their whippings and relief committees and houses of
correction, fresh, too, from enclosure and depopulation—complain
that their special subject is being overlooked in a general and](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-64-320.jpg)
![dangerous discussion on the economic causes of distress, and that
the committee “has spent all their travel about the said enclosures
and tillage, and nothing about the said rogues and poor.” That this
should have been the popular line to take needs no explanation. A
Parliament which dares discuss not only how to manipulate the lives
of the poor, but the fundamental causes of their misery, is a
Parliament which the eye of man had not yet, has not yet, beheld.
Compared with other representative assemblies, compared with itself
at a later date, the Elizabethan House of Commons, debating in an
age when it could be said that government was “nothing but a
certein conspiracy of riche men procuringe theire owne commodities
under the name and title of the Common Wealth,” had the grace to
show some stirrings of compunction. If members who had grown fat
on the tragedy which they were discussing spoke of their victims as
members will speak, ministers at least were independent, and could
venture, like Cecil, to tell the House unpalatable truths. Of the two
Acts against enclosure, which were the result of this session's
deliberations, we shall speak later. What is worth noticing here is the
disposition, even in a Parliament composed of country gentlemen, to
emphasise the connection between the problems with which anti-
enclosure and anti-vagrancy legislation have to deal. It is summed
up in the eloquent peroration of a nameless member. “As this bill
entered at first with a short prayer, 'God speed the plough,' so I wish
it may end with such success as the plough shall speed the
poor.”[477]
What became of the families displaced from the soil between their
final eviction and that subsidence upon the stony breast of the
Elizabethan Poor Law, which, for some of them, was their ultimate
fate? There is no certain information to guide us. The tragedy of the
tramp is his isolation. Every man’s hand is against him; and his
history is inevitably written by his enemies. Yet, beneath
denunciations hurled upon him by those who lived warm and slept
soft, we can see two movements going on, two waves in a vast and
silent ebbing of population from its accustomed seats. In the first
place there is a steady immigration into the towns on the part of](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-65-320.jpg)
![those “who, being driven out of their habitations, are forced into the
great cities, where, being very burdensome, men shut their doors
against them, suffering them to die in the streets and
highways.”[478] The municipal records of the periods teem with
complaints of the disorder, the overcrowding, the violation of
professional bye-laws, caused by rural immigration. The displaced
peasant is the Irishman of the sixteenth century, and, like the
Irishman, he makes his very misery a whip with which to scourge,
not alas! his oppressors, but men who often are not much less
wretched than himself. He turns whole quarters into slums, spreads
disease through congested town dwellings, and disorganises the
labour market by crowding out the native artisan. Gild members find
themselves eaten up by unlawful men who have never served an
apprenticeship in the town, and retort with regulations requiring the
deposit of a prohibitive sum as an entrance fee from all immigrants
who want to set up shop, especially from those wretches who are
thought to have a large family of children, at present snugly
concealed in their last place of residence, but soon to be
surreptitiously introduced, a brood of hungry young cuckoos, if once
their parents get a footing in the town.[479] Borough authorities, who
see cottages “made down" into tenements in which pestilence
spreads with fearful rapidity, seek to stamp out the very possibility of
invasion by prohibiting the erection of new cottages or the
subdivision of old. To judge by their behaviour, the notorious Statute
of 1662, which codified the existing customs as to settlement, must
have been one of the most popular pieces of legislation ever passed
by Parliament. Town[480] after town in the course of the sixteenth
century tries to protect itself by a system of stringent inspection
worthy of modern Germany. Sometimes there is a regular expulsion
of the aliens. “Forasmuch as it is found by daily experience,” declare
the authorities of Nottingham,[481] “that by the continual building
and erecting of new cottages and poor habitations, and by the
transferring of barns and suchlike buildings into cottages, and also
by the great confluence of many poor people from forrein parts out
of this towne to inhabit here, and lykewise by the usual and frequent
taking in of inmates into many poor habitations here, the poorer sort](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-66-320.jpg)
![of people do much increase ... it is ordered that no burgess or
freeman on pain of £5 erect any cottage or convert any building into
a cottage in the town without license of the Mayor, that no burgess
or freeman, without a license, receive any one from the country as a
tenant, that every landlord be bound in the sum of £10 to remove all
foreign tenants who have entered in the last three years before May
1st next.” What most boroughs do for themselves is finally, after
many regulations have been made by the Common Council, done for
London by Parliamentary legislation. It is not a chance that the end
of Elizabeth’s reign sees the first two Housing Acts, one[482] in 1589,
enacting that only one family may live in a house, the other[483]
applying to London alone, and forbidding the division of houses into
tenements, the receiving of lodgers, or the erection of new houses
for persons who are assessed in the subsidy book at less than £5 in
goods or £3 in lands. The evicted peasants are beginning to take
their revenge. They have been taking it ever since.
In the second place there is a general movement from the enclosed
to the open field villages. The families displaced by enclosure cannot
easily enter into industry, even if they wish to do so, for the avenue
to most trades is blocked both by the Corporations and by the
statutory system of a seven years' apprenticeship, which maintains
professional standards at the expense of an unprivileged residuum.
What they do is to follow the orthodox advice given to those who
have lost their customary means of livelihood. They proceed to
colonise, and to colonise in such numbers that they cannot easily be
kept out. They settle as squatters on the waste lands of those
manors which have not been enclosed, and which, before the waste
is turned into a sheep-run, offer no obstacle to immigration. That the
possibility of using the manorial waste to accommodate those who
had no settled abode had occurred to statesmen as one expedient
for meeting the problem of the infirm and destitute, is shown by the
sanction expressly given in the Poor Law of 1597[484] to the
expenditure of parish funds on the erection of cottages on the waste
as residences for the impotent poor. In fact, however, the mobility of
labour was becoming such that it was impossible, even if it had been](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-67-320.jpg)
![desirable, to reserve those unutilised territories for the maintenance
of the impotent. In spite of bitter protests from the existing
inhabitants, refugees from other villages swarm down upon them in
such numbers that the Act requiring four acres of land to be
attached to each cottage cannot be observed, and the issuing of
licences for the erection of cottages on the waste for able-bodied
men, who have come with their families from a distance, becomes a
regular part of the business of Quarter Sessions.[485] Such a
redistribution of the population solves one problem only to create
others. Stern economists in the seventeenth century lament that the
ease with which permission to build cottages on the waste is
obtained encourages the existence of an improvident and idle class,
which will neither work for wages nor make good use of the land. “In
all or most towns where the fields lie open and are used in common,
there is a new brood of upstart intruders as inmates, and the
inhabitants of unlawful cottages erected contrary unto law....
Loyterers who will not usually be got to work unless they may have
such excessive wages as they themselves desire.”[486] The
opponents of enclosure answer with some justice that, in effect, the
open field villages are saddled with the destitution caused by
enclosing landlords, who first ruin their tenants and then, like a
modern Dock Company which relies on the Poor Rate to save its
wage-bill, leave them to be supported by those places to which they
are compelled to migrate.[487] The latter difficulty is indeed a very
serious one, which not only is the occasion of numberless
petitions[488] from villages who wish to be assisted by, or to avoid
assisting, their neighbours, but on occasion converts even the
country gentry into opponents of enclosure. “We further conceive,”
write the Justices of Nottingham to the Council, “that if depopulation
may be reformed it will bring a great good to the whole Kingdom; for
where homes are pulled down the people are forced to seek new
habitations in other towns and countries, whereof those towns
where they get a settling are pestered so as they are hardly able to
live one by another, and it is likewise the cause of erecting new
cottages upon the waste and other places who are not able to
relieve themselves ... which causes rogues and vagabonds to](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-68-320.jpg)
![increase.”[489] In the elaborate book of Poor Law orders published in
1631 the Government recognises the genuineness of this grievance,
and, to its direction that richer parishes should contribute funds to
the aid of the poor, adds a special rider pointing out that such extra
contributions would come with special appropriateness from those
places where there had been depopulation.
We may now summarise our view of the social effects of the changes
introduced by lords of manors, and by the capitalist farmers who
manage their estates. When the demesne land is enclosed and
converted to pasture, there is an appreciable diminution in the
demand for labour, and consequently an increase in unemployment.
When the common rights of tenants are curtailed, they lose not only
an important subsidiary source of income, but often, at the same
time, the means of cultivating their arable holdings. When their
holdings are merged in the great estate of the capitalist farmer, they
are turned adrift to seek their living in a world where most trades
and most towns are barred against them, where they are punished if
they do not find work, and punished if they look for work without
permission, where “if the poor being thrust out of their houses go to
dwell with others, straight we catch them with the Statute of
Inmates; if they wander abroad, they are in danger of the Statute of
the Poor to be whipped.”[490] Thus, quite apart both from the eternal
source of poverty which consists in the recalcitrance of nature to
human effort, and from those causes of individual destitution which
in all ages and in all economic conditions lie in wait for the
exceptionally unfortunate or the exceptionally improvident, for the
sick, the aged, and the orphan, there is an increase in the number of
those for whom access to the land, their customary means of
livelihood, is unobtainable, and consequently a multiplication of the
residuum for whom the haunting insecurity of the propertyless
modern labourer is, not the exception, but the normal lot. It is this
extension of destitution among able-bodied men, who have the will,
but not the means, to find employment, which is the peculiar feature
of sixteenth century pauperism, and which leads in 1576 to the most
characteristic expedient of the Elizabethan Poor Law—the provision](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-69-320.jpg)
![of materials upon which the unemployed can be set to work. The
recognition that the relief of the destitute must be enforced as a
public obligation was not the consequence of the survival of
mediæval ideas into an age where they were out of place, but an
attempt on the part of the powerful Tudor state to prevent the social
disorder caused by economic changes, which, in spite of its efforts, it
had not been strong enough to control.[Next Chapter]
FOOTNOTES:
[416] The Shepe Book of Tittleshall Manor (Holkham MSS.,
Tittleshall Books, No. 19), shows flocks of 500 to 1000 sheep
being managed by a single shepherd, 1543–1549.
[417] e.g. Holkham MSS., Fulmordeston, Bdle. 6: “To the Right
Honourable Sir Edward Cooke, Knight, Attorney General unto the
King’s Matie. Humblie sheweth unto your lordship yor poore and
dayley orators ... yor worshippes tenants of the Manor of
Fulmordeston cum Croxton in the Duchie of Lancaster, and the
moste parte of the tenants of the same manor that whereas your
said orators in the Hillary Terme last commenced suite in the
Duchie Courte against Thomas Odbert and Roger Salisbury, gent.,
who have enclosed their grounds contrary to the custom of the
manor, wherby your wor. loseth your shack due out of the
grounds, common lane or way for passengers is stopped up, and
your worshipps' poore orators lose their accustomed shack in
those grounds, and the said Roger Salisbury taketh also the whole
benefit of theire common from them, keepinge there his sheepe in
grazinge, and debarring them of their libertie there which for
comon right belongeth unto them.” For the rest of this document
see Appendix I., and compare the following defence to a charge of
breaking open an enclosure: “The owners of the said tenements,
from time whereof there is no memory to the contrary, have had a
common of pasture for themselves and their tenants in one close
commonly called 'the new leasue,' in the lordship of Weston in the
manner following; that is to say, when the field where the said
'leasue' doth lie, called Radnor field, lieth fallow, then through the
whole year; and when the said field is sown with corn, then from
the reaping and carrying away of the corn until the same be sown
again ... and the said Thomas Dodd further said that he did break](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-70-320.jpg)
![open the said close ... being fenced in such time as he ought to
have common in the same, to the end that his cattle might take
their pasture therein" (William Salt Collection, New Series, vol. ix.,
Chancery Proceedings, Bdle. 8, No. 9).
[418] For complaints of tenants against the exactions, of farmers
as early as 1413, see Victoria County History, Essex, vol. ii. p.
318. For a stipulation in the farmer's covenant, see the following:
“Item a covenant conteyned in this lease that the said Thomas
shall permit and suffer the customary Tenants peaceably to have
and enjoy their estates, rights, grants, interests, and premises,
without any lette, interruption, or contradiction of the said
Thomas" (Roxburghe Club, Pembroke Surveys, Knyghton); and
Northumberland County History, vol. v. p. 208, Buston: “The
tenants of this town at the beginning of summer have their oxen
allway grazed in Shilbottel wood, or else they were not able to
maintain their tenements. It is therefore requisite that his lordship
or his heire should have respect unto the want of pasture, that in
any lease made by his lordship or his heire to any person of the
pasture, the said Shilbottel wood, there might be a proviso in the
said lease that the said tenants should have their oxen ground
there, as they have been accustomed.” Instances of the harrying
of the peasants by the large farmers are to be found, ibid., vol. i.
p. 350 (Tughall), and p. 274 (Newham).
[419] All Souls' Archives, vol. i. p. 203, No. 356.
[420] Topographer and Genealogist, vol. i., Survey of Mudford and
Hinton. In this case the aggressor was not the farmer of the
demesne, but a freeholder owning a third of the manor. To escape
his depredations the tenants proposed “to enclose their common
fieldes and to assign to Master Lyte and his tenants his third parte
in every field by itself, and to extinguish his right of common in
the rest.”
[421] Victoria County History, Suffolk, “Social and Economic
History.”
[422] For an amusing example see Conway, The Alps from End to
End, pp. 190–192.
[423] The Commonweal of this Realm of England, p. 57.
[424] Ten acres of “turf” to forty acres of arable was the estimate
of his requirements made to me by an Oxfordshire small holder.](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-71-320.jpg)
![[425] Topographer and Genealogist, vol. i.: “The tenants of
Landress have common in a certayne ground called King’s Moore
for all kinde of cattle, and every one of them may keep in the said
moore as much of all kind of cattle in somer as their severall or
ingrounde will beare in the wynter, whyche is a great relief to the
poore tenants, for as they confesse they keep all their cattle there
in the somer, and reserve their ingroundes untouched for the
winter.”
[426] e.g. Southampton Court Leet Records (Hearnshaw), pp. 4–
5, 1550: “Item we present that no burgers or comyners at one
time comyn above the number of two beasts upon payne of every
such defaulte 2s.; provided that iff any of them have two kyne or
wenlings, he shall have no horse, and yf he have but one cow he
may have one horse.”
[427] Topographer and Genealogist, vol. i.—Rolleston (Stafford):
“The said manor is ... well inhabited with divers honest men,
whose trade of lyvinge is onlie by husbandry ... and have no large
pastures or severall closes ... but have been alwaie accustomed to
have their cattle and sometyme their ploughe beasts pastured in
the Queen's Majestie’s Park of Rolleston, for xxd., the stage ...
without which aide and help they were neither able to maintain
hospitallitie nor tyllage; and nowe of late yeares the fermor of the
herbage hath advanced the stage to 6s. 8d., and yet the Quene’s
Majesties rent nothing increased.”
[428] Fitzherbert, Book of Husbandry.
[429] Northumberland County History, vol. v., Birling: “Allowed
part of 25s. 4d. for focage of Orchard Medow and Mylneside Bank,
because they are now enclosed within the lord’s new Park, and
this allowance shall be made yearly until the tenants of Byrling
have and peacefully enjoy another parcel of pasture to the same
value 11s. 8d." (Bailiff’s Accounts, 1474). R.O. Misc. Books Land
Rev., vol. ccxx., f. 236: “Divers parcels of land and pasture of the
manor of Farfield, now common of 140 acres, now occupied by
the tenants there as commons and given them in exchange in
satisfaction of their old common imparked in the new Park, £6,
13s. 8d.”
[430] Pollock and Maitland, History of English Law, vol. i. p. 606.
For the questions concerning common rights see ibid., pp. 594–
624, and Maitland, Domesday Book and Beyond, pp. 340–356;
Vinogradoff, Villainage in England, Essay II. chap, ii., and The](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-72-320.jpg)
![Growth of the Manor, Book II. chap. iv. I have followed
Vinogradoff’s rather than Maitland’s view.
[431] For buying and selling of pasture see below, and for
enclosure pp. 168–170. The following seems a clear case of more
or less corporate action. Holkham MSS., Burnham, Bdle. 5, No. 94:
“Copy of an indenture between [here follows a list of names] of
the same town and county, yeomen, as well on the behalf of
themselves as of the rest of the comoners and freeholders of the
said town of the one part, and Robert Bacon of [illegible] in the
County of Norfolk, and Thomas Coke of Grays Inn in the County of
Middlesex of the other part, that whereas heretofore Sir Philip
[illegible] being lord and owner of the marshes hereafter
mentioned ... did by his indenture of bargain and sale bearing
date ... 1588, grant bargain and sell unto [list of names as above]
all those marsh grounds lying and being in Burnham, to have and
to hold the said premises to the parties last before mentioned and
their heires to the use of them and their heires for ever, to the
intent and purpose notwithstanding that the said parties last
before mentioned there, being inhabitants in certain ancient
messuages in the said Towne, and all other inhabitants of the said
Towne there and afterwards for the tyme being in any of the
ancient messuages and cottages in the said towne, for so long
time as they shall be there inhabitinge and noe longer, according
to the quantity of their tenures within the said Towne might
depasture and feede the land as by the said deeds referring
thereunto being had may more fully appeare; [it recites that the
land] may by wallinge and embankinge the same be improved to
more than a [illegible] value, and made fitt for arrable, meadowe,
and pasture grounde, whereby tillage may be increased and his
Majestie’s subjects receive more employment thereby, and danger
of drawing [drowning?] of their stock for their feedinge prevented
[recites that Robert Bacon and Thomas Coke have undertaken to
drain the land in return for receiving three parts of it and that the
persons above mentioned] being the major parte of the parties
interested in the said salte Marshes, and being enabled by the
lawes and Statutes of this realm to contract and bargaine with any
person or persons for the draining thereof" [now convey 3 parts
of the marshes to the above-mentioned Robert Bacon and
Thomas Coke], June 8, 1637. The motive of this agreement was
to get the low-lying meadows on the sea-coast drained. Drainage
schemes were much in the air about this time, and any one who
has seen the country near Holkham and Burnham will know how](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-73-320.jpg)
![badly protection from the sea was needed. Two points are worth
noticing: (i.) the tenants have no objection to surrendering part of
their common if they get a quid pro quo; (ii.) they act as a single
body. They buy land and they sell land and they can leave it to
their heirs. Certain persons in the township act on their behalf,
much as directors might act for a body of shareholders. Is it
possible to speak of such arrangements simply in terms of
individual rights? Are we not driven to think of the township as
almost a landholding corporation?
[432] Common appendant, common appurtenant, common in
gross, and common par cause de vicinage. This classification is
not found in Bracton, and appears to date from the late Middle
Ages, see Vinogradoff, Villainage in England, Essay II., chap, ii.,
and the following case: Coke’s Reports, Part IV., p. 60. Hill, 4 Jac.
I. in Communi Banco: “Robert Smith brought an action of
Trespass against Stephen Gatewood, gent., quare clausum fregit
... cum quibusdam averiis.... Defendant pleaded a certain custom,
'quod inhabitantes infra eandem villam de Stixwood prædictam
infra aliquod antiquum messuagium ibidem ratione commorantiæ
et residentiæ suæ in eadem habuerunt et usi fuerunt et
consueverunt habere com. Pastur ... pro omnibus et omnimodis
bobus et equis et aliis grossis animalibus.' Unanimously resolved
that the custom is against law. 1. That there are but four manners
of common, common appendant, appurtenant, in gross, and by
reason of vicinage, and this common ratione commorantiæ is
none of them. 2. What estate shall he have, who is inhabitant, in
the common, when it appears he hath no estate or interest in the
house (but a mere habitation and dwelling) in respect of which he
ought to have his common? For none can have interest in a
common in respect of a house in which he hath no interest.”
[433] Coke, Complete Copyholder, Sect. 53: “When an Act of
Parliament altereth the service, tenure, or interest of the land, or
other thing in prejudice of the lord or of the Customs of the
Manor, or in prejudice of the tenant, then the generall words of
such an Act of Parliament extend not to the copyhold; but when
an Act is generally made for the good of the commonwealth, and
no prejudice may accrue by reason of the alteration of any
interest, service, tenure, or Custom, of the Manor, there usually
copyhold lands are within the generall purview of such Acts.”
[434] Fitzherbert, Book of Surveying: “And as for that manner of
common, me seemeth the Lord may improve himself of their](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-74-320.jpg)
![waste grounds, leaving their own tenants sufficient common,
having no regard to the tenants of the other lordship. But as far
as all errable lands, meadows, leises, and pastures, the lordes
may improve themselves by course of the common law, for the
statute speaketh nothing but of waste grounds.”
[435] e.g. Coventry Leet Book, vol. ii. p. 510.
[436] Genealoger and Archæologist, vol. i., Manor of West Coker
(Somerset): “The demesnes remayneth in one entier ferm, and is
dymysed to one Sir John Seymour, knight, who being confederate
with the freeholders of the manor, maketh such inclosers for his
owne lucre, and suffreth the freeholders to do the same,
nevertheless surcharge the common with their cattle, that in
process of tyme yt wilbe the destruccion of the custumarye
tenants.”
[437] For a discussion of the legal position of the copyholders see
below, pp. 287–310.
[438] Coventry Leet Book, vol. ii. pp. 445–446 and passim.
[439] If the common was so large that it had been unnecessary to
“stint” it, why did the city object to the lord putting additional
beasts on? I take the situation to be that the Prior—probably
tempted by the profitableness of sheep-farming in the latter part
of the fifteenth century—diminished the pasture which the city
could use, by putting on many more beasts than ever before,
which, in the absence of a recognised “stint,” he was able to do
without violating any custom, as he would have done if there had
been a customary limit, as on many manors.
[440] Topographer and Genealogist, vol. iii. These are the people
whom Heaven protected in the way described on p. 148 note.
Observe what this little community endured. (i.) Sir Francis
Englefield, senior, seizes 1900 out of 2000 acres of their common.
(ii.) Sir Francis Englefield, junior, seizes “the charter of our town ...
and the deed of the said common." (iii.) He tries to seize the
remaining 100 acres, and ruins them by lawsuits “for the space of
seven or eight years at the least, and never suffers any one to
come to triall in all that space ... that the said Free tenants were
not able to wage law any longer, for one John Rous ... was
thereby enforced to sell all his land (to the value of £500) with
following the suits in law, and many were thereby impoverished."
(iv.) He turns them out of their shops in the market-place, and](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-75-320.jpg)
![introduces instead “a stranger that liveth not in the town." (v.) He
appoints his own nominee as mayor, in defiance of the custom
which requires him to appoint one of two men submitted to him
by the jury. (vi.) He prevents his victims from signing this petition
by threats of eviction. ("They are fearful that they shall be put
forth of their bargaines, and then they shall not tell how to live,
otherwise they would have set to their hands.")
[441] Holkham MSS., Map of West Lexham.
[442] R.O. Aug. Off. Misc. Bks., vol. cccxcix., f. 201 ff.
[443] The manors are South Newton, Winterbourne Basset,
Knyghton, Donnington, and Estoverton and Phipheld (Roxburghe
Club, Surveys of Pembroke Manors).
[444] This, of course, is not inconsistent with a general
appreciation, i.e. a general rise in wages and fall in the rate of
interest.
[445] Northumberland County History, vol. ix. p. 124. For a similar
case of evictions by Delavale, showing how they were carried out,
ibid., pp. 201–202: “There was in Seaton Delavale township 12
tenements, whereon there dwelt 12 able men sufficiently
furnished with horse and furniture to serve his Majestie ... who
paid 46s. 8d. rent yearlie a piece or thereabouts. All the said
tenants and their successors saving 5 the said Robert Delavale
eyther thrust out of their fermholds or weried them by taking
excessive fines, increasing of their rents unto £3 a piece, and
withdrawing part of their best land and meadow from their
tenements ... by taking their good land from them and compelling
them to winne moorishe and heathe ground, and after their
hedging heth ground to their great charge, and paying a great
fine, and bestowing great reparation on building their tenements,
he quite thrust them off in one yeare, refusing either to repay the
fine or to repay the charge bestowed in diking or building.... The
said seven fermholds displaced had to every one of them 60 acres
of arable land, viz. 20 in every field at the least, as the tenants
affirme, which amounteth to 480 acres of land yearlie or
thereabouts, converted for the most part from tillage to pasture,
and united to the demaine of the lordship of Seaton Delavale.”
[446] In several cases the freeholders' lands are not stated in the
survey, and are therefore not included in this table.](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-76-320.jpg)
![[447] A few acres described as “held without title" are omitted.
[448] I am not sure that there are not other lands in Domerham
not included in the survey or in the demesne. If this is so, the
proportion of the latter to the rest of the manorial land would of
course be reduced.
[449] R.O. Rentals and Surveys, Gen. Ser., Portf. 22, No. 18.
[450] Roxburghe Club, Surveys of Pembroke Manors.
[451] Ibid., and Hoare, History of Wiltshire, Hundred of
Ambresbury.
[452] Northumberland County History, vol. i. p. 350.
[453] Ibid., vol. ix., Cowpen.
[454] Ibid., vol. i. p. 275.
[455] Ibid., vol. ix. pp. 201–202.
[456] Moore, The Crying Sin of England, &c.
[457] Cal. S. P. D. Eliz., 1595–1597 (p. 347), quoted Gay,
Quarterly Journal of Economics, vol. xvii.
[458] “Certayne Causes gathered together wherein is shewed the
decaye of England only by the great multitude of shepe" (E. E. T.
S. date 1550–1553). “It is to understande ... that there is in
England townes and villages to the number of fifty thousand and
upward, and for every town and village ... there is one plough
decayed since the fyrst year of the reign of King Henry VII.... The
whiche 50,000 ploughs every plough was able to maintain 6
persons, and nowe they have nothing, but goeth about in England
from dore to dore.”
[459] For a discussion of the value of these reports see Leadam,
Domesday of Enclosures, and Trans. Royal Hist. Soc., New Series,
vol. vi.; Gay, Trans. Royal Hist. Soc., New Series, vol. xiv. and vol.
xviii.; Gay, Quarterly Journal of Economics, vol. xvii. (1902–1903).
A useful summary of the evidence, with a map illustrating the
probable geographical distribution of the movement, is given by
Johnson, The Disappearance of the Small Landowner, pp. 42–54
and Map I.
[460] It is a question how far there had ever been an open field
system in some of these counties, e.g. Cornwall and Kent. There](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-77-320.jpg)
![certainly were some open field villages of the ordinary pattern in
Kent (see Slater, The English Peasantry and the Enclosure of
Common Fields, p. 230). But Kent from an early date develops on
its own lines, and does not go through the same stages of
manorialism and commutation as other counties. Much of it seems
to start at the point which they reach only in the sixteenth
century. Cornwall again, though in the sixteenth century there
were commons where the villagers pastured their cattle together
(see accounts of Landress and Porpehan, Topographer and
Genealogist, vol. i.), was largely a county of scattered homesteads
and very early enclosure (for the “nucleated village" and
“scattered homesteads,” see Maitland, Domesday Book and
Beyond, pp. 15–16), pointing to a different system of settlement
from that of the counties where the open field system obtained.
For enclosures in Devon and Somerset see Cunningham, Growth
of English Industry and Commerce, Modern Times, Part II., App.
B: “A consideration of the cause in question before the lords
touchinge depopulation," and Carlyle’s Cromwell, Letter XXIV.
“Lest we should engage our body of horse too far into that
enclosed country.”
[461] For intimidation see the case of Wootton Basset, quoted
above, pp. 251–253, and below, pp. 302–304. Also Gay, Trans.
Royal Hist. Soc., New Series, vol. xviii.; and Hales' defence
(appendix to Miss Lamond’s introduction to The Commonweal of
this Realm of England).
[462] Professor Pollard has good remarks on this point (Political
History of England, 1547–1603, p. 29).
[463] Wolsey was responsible for the Commission of 1517. For a
letter of Cromwell to Henry VIII. on the subject of enclosure, and
for the views of Cecil and Bacon, see below, pp. 273–274, 279,
343, 387.
[464] Mackay, History of the English Poor Law, 1834–1898, pp.
10–11, 16–17.
[465] Poor Law Commission Report of 1834, pp. 264–277, 281.
[466] 27 Hen. VIII., c. 25. Under this Act city and county
authorities are to relieve impotent beggars “by way of voluntary
and charitable alms.” They are also for the first time given power
to apprentice vagrant children.](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-78-320.jpg)
![[467] 18 Eliz. c. 3 directed that a stock of wool, flax, hemp, iron,
or other stuff should be provided in cities, corporate towns, and
market towns. The important words which show the change of
opinion are, “To the intente also that ... Roges ... may not have
any just excuse in saying they cannot get any service or work.”
[468] 14 Eliz. c. 5.
[469] Robert Cushman, “Reasons and Considerations touching the
Lawfulness of Removing out of England into the parts of America"
(printed by E. Arber, The Story of the Pilgrim Fathers).
[470] Harrison in Elizabethan England (Withington), chap. x.
[471] Hist. MSS. Com., Marquis of Salisbury, Part VII., pp. 160–
161: “Orders agreed to by the Justices of the Peace for Cornwall
at General Sessions for Bodmin the 5th and Truro the 8th of April,
39 Eliz.”
[472] Harrison, loc. cit.
[473] Ibid.
[474] Camden Society, 1886. Cases in Courts of Star Chamber and
High Commission, Michaelmas, 7 Caroli, Case of Archer. (The
allusion in the text is to a precedent cited in this case.)
[475] Hist. MSS. Com., Marquis of Salisbury, Part VII., Nov. 1597.
“Notes for the present Parliament.”
[476] D'Ewes' Journal, pp. 551–555; see also Leonard, The Early
History of English Poor Relief, pp. 73–75.
[477] Hist. MSS. Com., Marquis of Salisbury, Part VII., pp. 541–
543.
[478] Lansd. MSS. 83, f. 68, quoted Gonner, Common Land and
Enclosure, p. 156 n.
[479] e.g. Nottingham Records, vol. iv. pp. 170–171, Nov. 4,
1577: “Any burgess that hath not been prentice to pay £10 and no
pardon. Records of Leicester, vol. iii. p. 351, Oct. 17, 1598: “He is
inhibited from dwelling in your corporation unless he finds bonds
for £200 that neither his wife nor children shall be burdensome to
the town." Southampton Court Leet Records, vol. i., Part I.: “One
William Dye, undertenant to John Netley, dothe lyve idelly and
hathe no trade.... He hathe 4 or 5 children in places from whence](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-79-320.jpg)
![he came whom he will bring shortly hither, yf he may be suffered
here to remayne, whom we desyer may be examined and
removed from hence according to the Statute.”
[480] Some instances are given by Leonard, Early History of
English Poor Relief, pp. 107–109.
[481] Nottingham Records, vol. iv. pp. 304–307.
[482] 31 Eliz. c. 7.
[483] 35 Eliz. c. 6.
[484] 39 Eliz. c. 3.
[485] For petitions on this subject see Hist. MSS. Com., Cd. 784,
pp. 81–82 (Wiltshire). The Warwickshire Quarter Sessions were
much occupied with this, e.g. the following: “Trinity Sessions
1625. Fforasmuch as this Court was this present day informed ...
by Sir Edward Marrowe, kt., and Thomas Ashley as the lords of
the manor of Woolvey in this county ... that the said lords are
content that William Wilcox of Woolvey in this countie shall build
and erect a cottage for hys habitation hys wyfe and his small
children uppon the waste within the said lordshippe, it is therefore
ordered that the same being with consent of the lord as aforesaid
that the same cottage shall be and continue,” and later “which
cottage the Court doth licence" (Warwick Quarter Sessions MSS.
Records).
[486] “Considerations Concerning Common Fields and Enclosures,”
Pseudonismus, 1654.
[487] Moore, The Crying Sin of England in not Caring for the Poor:
“And now alas, saith the poor cottier, there is no work for me, I
must go where I may get my living. And hence it comes to pass
that the open fielden towns have above double the number of
cottiers they had wont to have, so that they cannot live one by
another, and so put the fielden towns to vast expense, in caring
for these poor that these enclosures have made.”
[488] e.g. Hist. MSS. Com., Cd. 784, p. 95 (Wiltshire), pp. 292
and 298 (Worcester).
[489] See Appendix I., No. VI. Miss Leonard (Trans. Royal Hist.
Soc., vol. xix.) prints this document as referring to Norfolk, which
appears to be an error.](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-80-320.jpg)
![[490] D'Ewes' Journal. Speech of Cecil, 1597.](https://image.slidesharecdn.com/465642-250520144555-7f0909b8/85/Structure-And-Interpretation-Of-Classical-Mechanics-Gerald-Jay-Sussman-81-320.jpg)
Structure And Interpretation Of Classical Mechanics Gerald Jay Sussman
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- 5. Structure and Interpretation of Classical Mechanics
- 7. Structure and Interpretation of Classical Mechanics Gerald Jay Sussman and Jack Wisdom with Meinhard E. Mayer The MIT Press Cambridge, Massachusetts London, England
- 8. c °2000 by The Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. This book was set by the authors using the L A TEX typesetting system and was printed and bound in the United States of America.
- 9. This book is dedicated, in respect and admiration, to The Principle of Least Action. “The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated. In the interest of clearness, it appeared to me inevitable that I should repeat myself frequently, without pay- ing the slightest attention to the elegance of the presentation. I adhered scrupulously to the precept of that brilliant theoretical physicist L. Boltzmann, according to whom matters of elegance ought be left to the tailor and to the cobbler.” Albert Einstein, in Relativity, the Special and General Theory, (1961), p. v.
- 11. Contents Contents vii Preface xiii Acknowledgments xvii 1 Lagrangian Mechanics 1 1.1 The Principle of Stationary Action 4 1.2 Configuration Spaces 9 1.3 Generalized Coordinates 11 1.4 Computing Actions 16 1.5 The Euler-Lagrange Equations 26 1.5.1 Derivation of the Lagrange Equations 27 1.5.2 Computing Lagrange’s Equations 34 1.6 How to Find Lagrangians 37 1.6.1 Coordinate Transformations 44 1.6.2 Systems with Rigid Constraints 48 1.6.3 Constraints as Coordinate Transformations 60 1.6.4 The Lagrangian is Not Unique 62 1.7 Evolution of Dynamical State 67 1.8 Conserved Quantities 76 1.8.1 Conserved Momenta 76 1.8.2 Energy Conservation 78 1.8.3 Central Forces in Three Dimensions 81 1.8.4 Noether’s Theorem 84 1.9 Abstraction of Path Functions 88 1.10 Constrained Motion 93 1.10.1Coordinate Constraints 95
- 12. viii Contents 1.10.2Derivative Constraints 102 1.10.3Non-Holonomic Systems 106 1.11 Summary 109 1.12 Projects 110 2 Rigid Bodies 113 2.1 Rotational Kinetic Energy 114 2.2 Kinematics of Rotation 116 2.3 Moments of Inertia 118 2.4 Inertia Tensor 121 2.5 Principal Moments of Inertia 123 2.6 Representation of the Angular Velocity Vector 125 2.7 Euler Angles 128 2.8 Vector Angular Momentum 132 2.9 Motion of a Free Rigid Body 134 2.9.1 Computing the Motion of Free Rigid Bodies 136 2.9.2 Qualitative Features 138 2.10 Axisymmetric Tops 144 2.11 Spin-Orbit Coupling 152 2.11.1Development of the Potential Energy 152 2.11.2Rotation of the Moon and Hyperion 156 2.12 Euler’s Equations 163 2.13 Nonsingular Generalized Coordinates 168 2.14 Summary 177 2.15 Projects 177 3 Hamiltonian Mechanics 179 3.1 Hamilton’s Equations 181 3.1.1 The Legendre Transformation 189 3.1.2 Hamiltonian Action Principle 199 3.1.3 A Wiring Diagram 201 3.2 Poisson Brackets 203
- 13. Contents ix 3.3 One Degree of Freedom 206 3.4 Phase Space Reduction 208 3.4.1 Lagrangian Reduction 218 3.5 Phase Space Evolution 220 3.5.1 Phase Space Description is Not Unique 222 3.6 Surfaces of Section 224 3.6.1 Poincaré Sections for Periodically-Driven Systems 225 3.6.2 Computing Stroboscopic Surfaces of Section 231 3.6.3 Poincaré Sections for Autonomous Systems 232 3.6.4 Non-axisymmetric Top 245 3.7 Exponential Divergence 246 3.8 Liouville’s Theorem 250 3.9 Standard Map 259 3.10 Summary 262 3.11 Projects 263 4 Phase Space Structure 265 4.1 Emergence of the Mixed Phase Space 266 4.2 Linear Stability of Fixed Points 271 4.2.1 Equilibria of Differential Equations 271 4.2.2 Fixed Points of Maps 275 4.2.3 Relations Among Exponents 277 4.3 Homoclinic Tangle 282 4.3.1 Computation of Stable and Unstable Manifolds 287 4.4 Integrable Systems 289 4.5 Poincaré-Birkhoff Theorem 296 4.5.1 Computing the Poincaré-Birkhoff Construction 301 4.6 Invariant Curves 303 4.6.1 Finding Invariant Curves 306 4.6.2 Dissolution of Invariant Curves 311 5 Canonical Transformations 317 5.1 Point Transformations 318 5.2 General Canonical Transformations 322
- 14. x Contents 5.2.1 Time-independent Canonical Transformations 325 5.2.2 Symplectic Transformations 330 5.2.3 Time-Dependent Transformations 333 5.2.4 The Symplectic Condition 336 5.3 Invariants of Canonical Transformations 338 5.4 Extended Phase Space 345 5.4.1 Poincaré-Cartan Integral Invariant 352 5.5 Reduced Phase Space 353 5.6 Generating Functions 358 5.6.1 F1 Generates Canonical Transformations 360 5.6.2 Generating Functions and Integral Invariants 362 5.6.3 Classes of Generating Functions 368 5.6.4 Point Transformations 370 5.6.5 Classical “Gauge” Transformations 386 5.7 Time Evolution is Canonical 391 5.7.1 Another View of Time Evolution 397 5.7.2 Yet Another View of Time Evolution 401 5.8 Hamilton-Jacobi Equation 403 5.8.1 Harmonic Oscillator 405 5.8.2 Kepler Problem 409 5.8.3 F2 and the Lagrangian 413 5.8.4 The Action Generates Time Evolution 414 5.9 Lie Transforms 416 5.10 Lie Series 422 5.11 Exponential Identities 430 5.12 Summary 432 6 Canonical Perturbation Theory 435 6.1 Perturbation Theory with Lie Series 436 6.2 Pendulum as a Perturbed Rotor 438 6.2.1 Higher Order 446 6.2.2 Eliminating Secular Terms 448 6.3 Many Degrees of Freedom 451 6.3.1 Driven Pendulum as a Perturbed Rotor 454
- 15. Contents xi 6.4 Nonlinear Resonance 456 6.4.1 Pendulum Approximation 458 6.4.2 Reading the Hamiltonian 464 6.4.3 Resonance Overlap Criterion 466 6.4.4 Resonances in Higher Order Perturbation Theory 467 6.4.5 Stability of Inverted Vertical Equilibrium 468 6.5 Projects 472 7 Appendix: Our Notation 475 8 Appendix: Scheme 491 Bibliography 501 List of Exercises 505
- 17. Preface “In almost all textbooks, even the best, this principle is presented so that it is impossible to understand.” (K. Jacobi Lectures on Dynamics, 1842-1843). I have not chosen to break with tradition. V.I. Arnold, Mathematical Methods of Classical Mechanics (1980), footnote on p. 246 There has been a remarkable revival of interest in classical me- chanics in recent years. We now know that there is much more to classical mechanics than previously suspected. The behavior of classical systems is surprisingly rich; derivation of the equations of motion, the focus of traditional presentations of mechanics, is just the beginning. Classical systems display a complicated array of phenomena such as non-linear resonances, chaotic behavior, and transitions to chaos. Traditional treatments of mechanics concentrate most of their effort on the extremely small class of symbolically tractable dy- namical systems. We concentrate on developing general methods for studying the behavior of systems, whether or not they have a symbolic solution. Typical systems exhibit behavior that is qualitatively different from the solvable systems and surprisingly complicated. We focus on the phenomena of motion, and we make extensive use of computer simulation to explore this motion. Even when a system is not symbolically tractable the tools of modern dynamics allow one to extract a qualitative understand- ing. Rather than concentrating on symbolic descriptions, we con- centrate on geometric features of the set of possible trajectories. Such tools provide a basis for the systematic analysis of numerical or experimental data. Classical mechanics is deceptively simple. It is surprisingly easy to get the right answer with fallacious reasoning or without real understanding. Traditional mathematical notation contributes to this problem. Symbols have ambiguous meanings, which de-
- 18. xiv Preface pend on context, and often even change within a given context.1 For example, a fundamental result of mechanics is the Lagrange equations. Using traditional notation the Lagrange equations are written d dt ∂L ∂q̇i − ∂L ∂qi = 0. The Lagrangian L must be interpreted as a function of the position and velocity components qi and q̇i, so that the partial deriva- tives make sense, but then in order for the time derivative d/dt to make sense solution paths must have been inserted into the partial derivatives of the Lagrangian to make functions of time. The traditional use of ambiguous notation is convenient in simple situations, but in more complicated situations it can be a serious handicap to clear reasoning. In order that the reasoning be clear and unambiguous, we have adopted a more precise mathematical notation. Our notation is functional and follows that of modern mathematical presentations.2 Computation also enters into the presentation of the mathe- matical ideas underlying mechanics. We require that our mathe- matical notations be explicit and precise enough so that they can 1 In his book on mathematical pedagogy [15], Hans Freudenthal argues that the reliance on ambiguous, unstated notational conventions in such expressions as f(x) and df(x)/dx makes mathematics, and especially introductory calcu- lus, extremely confusing for beginning students; and he enjoins mathematics educators to use more formal modern notation. 2 In his beautiful book Calculus on Manifolds (1965), Michael Spivak uses functional notation. On p.44 he discusses some of the problems with classical notation. We excerpt a particularly juicy quote: The mere statement of [the chain rule] in classical notation requires the introduction of irrelevant letters. The usual evaluation for D1(f ◦(g, h)) runs as follows: If f(u, v) is a function and u = g(x, y) and v = h(x, y) then ∂f(g(x, y), h(x, y)) ∂x = ∂f(u, v) ∂u ∂u ∂x + ∂f(u, v) ∂v ∂v ∂x [The symbol ∂u/∂x means ∂/∂x g(x, y), and ∂/∂u f(u, v) means D1f(u, v) = D1f(g(x, y), h(x, y)).] This equation is often written simply ∂f ∂x = ∂f ∂u ∂u ∂x + ∂f ∂v ∂v ∂x . Note that f means something different on the two sides of the equation!
- 19. Preface xv be interpreted automatically, as by a computer. As a consequence of this requirement the formulas and equations that appear in the text stand on their own. They have clear meaning, independent of the informal context. For example, we write Lagrange’s equations in functional notation as follows:3 D(∂2L ◦ Γ[q]) − ∂1L ◦ Γ[q] = 0 The Lagrangian L is a real-valued function of time t, coordinates x, and velocities v; the value is L(t, x, v). Partial derivatives are indicated as derivatives of functions with respect to partic- ular argument positions; ∂2L indicates the function obtained by taking the partial derivative of the Lagrangian function L with respect to the velocity argument position. The traditional partial derivative notation, which employs a derivative with respect to a “variable,” depends on context and can lead to ambiguity.4 The partial derivatives of the Lagrangian are then explicitly evaluated along a path function q. The time derivative is taken and the Lagrange equations formed. Each step is explicit; there are no implicit substitutions. Computational algorithms are used to communicate precisely some of the methods used in the analysis of dynamical phenomena. Expressing the methods of variational mechanics in a computer language forces them to be unambiguous and computationally effective. Computation requires us to be precise about the repre- sentation of mechanical and geometric notions as computational objects and permits us to represent explicitly the algorithms for manipulating these objects. Also, once formalized as a procedure, a mathematical idea becomes a tool that can be used directly to compute results. Active exploration on the part of the student is an essential part of the learning experience. Our focus is on understanding the motion of systems; to learn about motion the student must actively explore the motion of systems through simulation and 3 This is presented here without explanation, to give the flavor of the notation. The text gives a full explanation. 4 “It is necessary to use the apparatus of partial derivatives, in which even the notation is ambiguous.” From V.I. Arnold, Mathematical Methods of Classical Mechanics (1980), Section 47, p258. See also the footnote on that page.
- 20. xvi Preface experiment. The exercises and projects are an integral part of the presentation. That the mathematics is precise enough to be interpreted au- tomatically allows active exploration to be extended to the math- ematics. The requirement that the computer be able to inter- pret any expression provides strict and immediate feedback as to whether the expression is correctly formulated. Experience demonstrates that interaction with the computer in this way un- covers and corrects many deficiencies in understanding. This book presents classical mechanics from an unusual per- spective. It focuses on understanding motion rather than deriving equations of motion. It weaves recent discoveries of nonlinear dy- namics throughout the presentation, rather than presenting them as an afterthought. It uses functional mathematical notation that allows precise understanding of fundamental properties of classical mechanics. It uses computation to constrain notation, to capture and formalize methods, for simulation, and for symbolic analysis. This book is the result of teaching classical mechanics at MIT for the past six years. The contents of our class began with ideas from a class on nonlinear dynamics and solar system dynamics by Wisdom and ideas about how computation can be used to formu- late methodology developed in the introductory computer science class by Abelson and Sussman. When we started we expected that using this approach to formulate mechanics would be easy. We quickly learned though that there were many things we thought we understood that we did not in fact understand. Our requirement that our mathematical notations be explicit and precise enough so that they can be interpreted automatically, as by a computer, is very effective in uncovering puns and flaws in reasoning. The resulting struggle to make the mathematics precise, yet clear and computationally effective, lasted far longer than we anticipated. We learned a great deal about both mechanics and computation by this process. We hope others, especially our competitors, will adopt these methods that enhance understanding, while slowing research.
- 21. Acknowledgments We would like to thank the many people who have helped us to develop this book and the curriculum that it is designed to sup- port. We have had substantial help from the wonderful students who studied with us in our classical mechanics class. They have forced us to be clear; they have found bugs that we had to fix, in the software, in the presentation, and in our thinking. We have had considerable technical help in the development and presentation of the subject matter from Harold Abelson. Abelson is one of the developers of the Scmutils software system. He put mighty effort into some sections of the code. We also consulted him when we were desperately trying to understand the logic of mechanics. He often could propose a direction to lead out of an intellectual maze. Matthew Halfant started us on the development of the Scmutils system. He encouraged us to get into scientific computation, using Scheme and functional style as an active way to explain the ideas, without the distractions of imperative languages such as C. In the 1980’s he wrote some of the early Scheme procedures for numerical computation that we still use. Dan Zuras helped us with the invention of the unique organi- zation of the Scmutils system. It is because of his insight that the system is organized around a generic extension of the chain rule for taking derivatives. He also helped in the heavy lifting that was required to make a really good polynomial GCD algorithm, based on ideas that we learned from Richard Zippel. This book, and a great deal of other work of our laboratory, could not have been done without the outstanding work of Chris Hanson. Chris developed and maintained the Scheme system un- derlying this work. In addition, he took us through a pass of reorganization of the Scmutils system that forced the clarification of many of the ideas of types and of generic operations that make our system as good as it is. Guillermo Juan Rozas, co-developer of the Scheme system, made major contributions to the Scheme compiler, and imple-
- 22. xviii Acknowledgments mented a number of other arcane mechanisms that make our system efficient enough to support our work. Besides contributing to some of the methods for the solution of linear equations in the Scmutils system, Jacob Katzenelson has provided valuable feedback that improved the presentation of the material. Julie Sussman, PPA, provided careful reading and serious crit- icism that forced us to reorganize and rewrite major parts of the text. She also developed and maintained Gerald Jay Sussman over these many years. Cecile Wisdom, saint, is a constant reminder, by her faith and example, of what is really important. This project would not have been possible without the loving support and unfailing en- couragement she has given Jack Wisdom. Their children, William, Edward, Thomas, John, and Elizabeth Wisdom, each wonderfully created, daily enrich his life with theirs. Meinhard Mayer wants to thank Rita Mayer, for patient moral support, particularly during his frequent visits to MIT during the last 12 years; Niels Mayer for introducing him to the wonderful world of Scheme (thus sowing the seeds for this collaboration), as well as Elma and the rest of the family for their love. Many have contributed to our understanding of dynamics over the years. Michel Henon and Boris Chirikov have had particular influence. Stan Peale, Peter Goldreich, Alar Toomre, and Scott Tremaine have been friends and mentors. We thank former stu- dents Jihad Touma and Matthew Holman, for their collaboration and continued friendship. We have greatly benefited from associ- ations with many in the nonlinear dynamics community: Tassos Bountis, Robert Helleman, Michael Berry, Michael Tabor, Ian Percival, John Greene, Robert MacKay, Jim Meiss, Dominique Escande, David Ruelle, Mitchell Feigenbaum, Leo Kadanoff, Jim Yorke, Celso Grebogi, Steve Wiggins, Philip Holmes, Jerry Gollub, Harry Swinney, and many others. We also acknowledge the late Res Jost, George Sudarshan, and Walter Thirring. There are numerous others who have contributed to this work, either in the development of the software or in the development of the content, including Bill Siebert, Panayotis Skordos, Kleanthes Koniaris, Kevin Lin, James McBride, Rebecca Frankel, Thomas F. Knight, Pawan Kumar, Elizabeth Bradley, Alice Seckel, and Kenneth Yip. We have had extremely useful feedback from and
- 23. Acknowledgments xix discussions with Piet Hut, Jon Doyle, David Finkelstein, Peter Fisher, and Robert Hermann. We thank The MIT Artificial Intelligence Laboratory for its hos- pitality and logistical support. We acknowledge the Matsushita Corporation for support of Gerald Jay Sussman through an en- dowed chair. We thank Breene M. Kerr for support of Jack Wis- dom through an endowed chair. We thank the MIT Mathemat- ics and EECS departments for sabbatical support for Meinhard Mayer. And finally, we are grateful to Rebecca Bisbee for her assistance over the many years we have been involved in this project.
- 25. Structure and Interpretation of Classical Mechanics
- 27. 1 Lagrangian Mechanics The purpose of mechanics is to describe how bodies change their position in space with “time.” I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations. Let us proceed to disclose these sins. Albert Einstein Relativity, the Special and General Theory, (1961), p. 9. The subject of this book is motion, and the mathematical tools used to describe it. Centuries of careful observations of the motions of the planets revealed regularities in those motions, allowing accurate predic- tions of phenomena such as eclipses and conjunctions. The effort to formulate these regularities and ultimately to understand them led to the development of mathematics and to the discovery that mathematics could be effectively used to describe aspects of the physical world. That mathematics can be used to describe natural phenomena is a remarkable fact. When a juggler throws a pin it takes a rather predictable path and it rotates in a rather predictable way. In fact, the skill of jug- gling depends crucially on this predictability. It is also a remark- able discovery that the same mathematical tools used to describe the motions of the planets can be used to describe the motion of the juggling pin. Classical mechanics describes the motion of a system of par- ticles, subject to forces describing their interactions. Complex physical objects, such as juggling pins, can be modeled as myriad particles with fixed spatial relationships maintained by stiff forces of interaction. There are many conceivable ways a system could move that never occur. We can imagine that the juggling pin might pause in midair or go fourteen times around the head of the juggler be- fore being caught, but these motions do not happen. How can
- 28. 2 Chapter 1 Lagrangian Mechanics we distinguish motions of a system that can actually occur from other conceivable motions? Perhaps we can invent some mathe- matical function that allows us to distinguish realizable motions from among all conceivable motions. The motion of a system can be described by giving the position of every piece of the system at each moment. Such a description of the motion of the system is called a configuration path; the config- uration path specifies the configuration as a function of time. The juggling pin rotates as it flies through the air; the configuration of the juggling pin is specified by giving the position and orientation of the pin. The motion of the juggling pin is specified by giving the position and orientation of the pin as a function of time. The function that we seek takes a configuration path as an input and produces some output. We want this function to have some characteristic behavior when the input is a realizable path. For example, the output could be a number, and we could try to arrange that the number is zero only on realizable paths. Newton’s equations of motion are of this form; at each moment Newton’s differential equations must be satisfied. However, there is a alternate strategy that provides more in- sight and power: we could look for a path-distinguishing function that has a minimum on the realizable paths—on nearby unreal- izable paths the value of the function is higher than it is on the realizable path. This is the variational strategy: for each physical system we invent a path-distinguishing function that distinguishes realizable motions of the system by having a stationary point for each realizable path.1 For a great variety of systems realizable motions of the system can be formulated in terms of a variational principle.2 1 A stationary point of a function is a point where the function’s value does not vary as the input is varied. Local maxima or minima are stationary points. 2 The variational formulation successfully describes all of the Newtonian me- chanics of particles and rigid bodies. The variational formulation has also been usefully applied in the description of many other systems such as classi- cal electrodynamics, the dynamics of inviscid fluids, and the design of mech- anisms such as four-bar linkages. In addition, modern formulations of quan- tum mechanics and quantum field theory build on many of the same con- cepts. However, the variational formulation does not appear to apply to all dynamical systems. For example, there is no simple prescription to apply the variational apparatus to systems with dissipation, though in special cases variational methods still apply.
- 29. 3 Mechanics, as invented by Newton and his contemporaries, de- scribes the motion of a system in terms of the positions, velocities, and accelerations of each of the particles in the system. In contrast to the Newtonian formulation of mechanics, the variational formu- lation of mechanics describes the motion of a system in terms of aggregate quantities that are associated with the motion of the system as a whole. In the Newtonian formulation the forces can often be written as derivatives of the potential energy of the system. The motion of the system is determined by considering how the individual component particles respond to these forces. The Newtonian for- mulation of the equations of motion is intrinsically a particle-by- particle description. In the variational formulation the equations of motion are for- mulated in terms of the difference of the kinetic energy and the potential energy. The potential energy is a number that is char- acteristic of the arrangement of the particles in the system; the kinetic energy is a number that is determined by the velocities of the particles in the system. Neither the potential energy nor the kinetic energy depend on how those positions and velocities are specified. The difference is characteristic of the system as a whole and does not depend on the details of how the system is specified. So we are free to choose ways of describing the system that are easy to work with; we are liberated from the particle-by-particle description inherent in the Newtonian formulation. The variational formulation has numerous advantages over the Newtonian formulation. The equations of motion for those param- eters that describe the state of the system are derived in the same way regardless of the choice of those parameters: the method of formulation does not depend on the choice of coordinate system. If there are positional constraints among the particles of a system the Newtonian formulation requires that we consider the forces maintaining these constraints, whereas in the variational formu- lation the constraints can be built into the coordinates. The vari- ational formulation reveals the association of conservation laws with symmetries. The variational formulation provides a frame- work for placing any particular motion of a system in the context of all possible motions of the system. We pursue the variational formulation because of these advantages.
- 30. 4 Chapter 1 Lagrangian Mechanics 1.1 The Principle of Stationary Action Let us suppose that for each physical system there is a path- distinguishing function that is stationary on realizable paths. We will try to deduce some of its properties. Experience of motion Our ordinary experience suggests that physical motion can be de- scribed by configuration paths that are continuous and smooth.3 We do not see the juggling pin jump from one place to another. Nor do we see the juggling pin suddenly change the way it is mov- ing. Our ordinary experience suggests that the motion of physical systems does not depend upon the entire history of the system. If we enter the room after the juggling pin has been thrown into the air we cannot tell when it left the juggler’s hand. The juggler could have thrown the pin from a variety of places at a variety of times with the same apparent result as we walk in the door.4 So the motion of the pin does not depend on the details of the history. Our ordinary experience suggests that the motion of physical systems is deterministic. In fact, a small number of parameters summarize the important aspects of the history of the system and determine the future evolution of the system. For example, at any moment the position, velocity, orientation and rate of change of the orientation of the juggling pin are enough to completely determine the future motion of the pin. Realizable paths From our experience of motion we develop certain expectations about realizable configuration paths. If a path is realizable, then any segment of the path is a realizable path segment. Conversely, a path is realizable if every segment of the path is a realizable 3 Experience with systems on an atomic scale suggests that at this scale systems do not travel along well-defined configuration paths. To describe the evolution of systems on the atomic scale we employ quantum mechanics. Here, we restrict attention to systems for which the motion is well described by a smooth configuration path. 4 Extrapolation of the orbit of the Moon backward in time cannot determine the point at which the Moon was placed on this trajectory. To determine the origin of the Moon we must supplement dynamical evidence with other physical evidence such as chemical compositions.
- 31. 1.1 The Principle of Stationary Action 5 path segment. The realizability of a path segment depends on all points of the path in the segment. The realizability of a path segment depends on every point of the path segment in the same way; no part of the path is special. The realizability of a path segment depends only on points of the path within the segment; the realizability of a path segment is a local property. So the path-distinguishing function aggregates some local prop- erty of the system measured at each moment along the path seg- ment. Each moment along the path must be treated the same way. The contributions from each moment along the path segment must be combined in a way that maintains the independence of the con- tributions from disjoint subsegments. One method of combination that satisfies these requirements is to add up the contributions, making the path-distinguishing function an integral over the path segment of some local property of the path.5 So we will try to arrange that the path-distinguishing func- tion, constructed as an integral of a local property along the path, assumes an extreme value for any realizable path. Such a path- distinguishing function is traditionally called an action for the system. We use the word “action” to be consistent with common usage. Perhaps it would be clearer to continue to call it “path- distinguishing function,” but then it would be more difficult for others to know what we were talking about.6 In order to pursue the agenda of variational mechanics, we must invent action functions that are stationary on the realizable tra- jectories of the systems we are studying. We will consider actions that are integrals of some local property of the configuration path at each moment. Let γ be the configuration-path function; γ(t) 5 We suspect that this argument can be promoted to a precise constraint on the possible ways of making this path-distinguishing function. 6 Historically, Huygens was the first to use the term “action” in mechanics. He used the term to refer to “the effect of a motion.” This is an idea that came from the Greeks. In his manuscript “Dynamica” (1690) Leibnitz enunciated a “Least Action Principle” using the “harmless action,” which was the product of mass, velocity, and the distance of the motion. Leibnitz also spoke of a “violent action” in the case where things collided.
- 32. 6 Chapter 1 Lagrangian Mechanics is the configuration at time t. The action of the segment of the path γ in the time interval from t1 to t2 is7 S[γ](t1, t2) = Z t2 t1 F[γ] (1.1) where F[γ] is a function of time that measures some local property of the path. It may depend upon the value of the function γ at that time and the value of any derivatives of γ at that time.8 The configuration path can be locally described at a moment in terms of the configuration, the rate of change of the configuration, and all the higher derivatives of the configuration at the given moment. Given this information the path can be reconstructed in some interval containing that moment.9 Local properties of paths can depend on no more than the local description of the path. The function F measures some local property of the configura- tion path γ. We can decompose F[γ] into two parts: a part that measures some property of a local description and a part that ex- tracts a local description of the path from the path function. The function that measures the local property of the system depends on the particular physical system; the method of construction of a local description of a path from a path is the same for any system. We can write F[γ] as a composition of these two functions:10 F[γ] = L ◦ T [γ]. (1.2) 7 A definite integral of a real-valued function f of a real argument is written R b a f. This can also be written R b a f(x)dx. The first notation emphasizes that a function is being integrated. 8 Traditionally, square brackets are put around functional arguments. In this case, the square brackets remind us that the value of S may depend on the function γ in complicated ways, such as through its derivatives. 9 In the case of a real-valued function the value of the function and its deriva- tives at some point can be used to construct a power series. For sufficiently nice functions (real analytic) the power series constructed in this way con- verges in some interval containing the point. Not all functions can be locally represented in this way. For example, the function f(x) = exp(−1/x2 ), with f(0) = 0, is zero and has all derivatives zero at x = 0, but this infinite number of derivatives is insufficient to determine the function value at any other point. 10 Here ◦ denotes composition of functions: (f◦g)(t) = f(g(t)). In our notation the application of a path-dependent function to its path is of higher precedence than the composition, so L ◦ T [γ] = L ◦ (T [γ]).
- 33. 1.1 The Principle of Stationary Action 7 The function T takes the path and produces a function of time. Its value is an ordered tuple containing the time, the configuration at that time, the rate of change of the configuration at that time, and the values of higher derivatives of the path evaluated at that time. For the path γ and time t:11 T [γ](t) = (t, γ(t), Dγ(t), . . .) (1.3) We refer to this tuple, which includes as many derivatives as are needed, as the local tuple. The function L depends on the specific details of the physical system being investigated, but does not depend on any particular configuration path. The function L computes a real-valued local property of the path. We will find that L needs only a finite num- ber of components of the local tuple to compute this property: The path can be locally reconstructed from the full local descrip- tion; that L depends on a finite number of components of the local tuple guarantees that it measures a local property.12 The advantage of this decomposition is that the local descrip- tion of the path is computed by a uniform process from the con- figuration path, independent of the system being considered. All of the system-specific information is captured in the function L. The function L is called a Lagrangian13 for the system, and the resulting action, S[γ](t1, t2) = Z t2 t1 L ◦ T [γ], (1.4) 11 The derivative Dγ of a configuration path γ can be defined in terms of ordinary derivatives by specifying how it acts on sufficiently smooth real- valued functions f of configurations. The exact definition is unimportant at this stage. If you are curious see footnote 23. 12 We will later discover that an initial segment of the local tuple will be sufficient to determine the future evolution of the system. That a configuration and a finite number of derivatives determines the future means that there is a way of determining all of the rest of the derivatives of the path from the initial segment. 13 The classical Lagrangian plays a fundamental role in the path-integral for- mulation of quantum mechanics (due to Dirac and Feynman), where the com- plex exponential of the classical action yields the relative probability ampli- tude for a path. The Lagrangian is the starting point for the Hamiltonian formulation of mechanics (discussed in chapter 3), which is also essential in the Schrödinger and Heisenberg formulations of quantum mechanics and in the Boltzmann-Gibbs approach to statistical mechanics.
- 34. 8 Chapter 1 Lagrangian Mechanics is called the Lagrangian action. Lagrangians can be found for a great variety of systems. We will see that for many systems the Lagrangian can be taken to be the difference between kinetic and potential energy. Such Lagrangians depend only on the time, the configuration, and the rate of change of the configuration. We will focus on this class of systems, but will also consider more general systems from time to time. A realizable path of the system is to be distinguished from oth- ers by having stationary action with respect to some set of nearby unrealizable paths. Now some paths near realizable paths will also be realizable: for any motion of the juggling pin there is an- other that is slightly different. So when addressing the question of whether the action is stationary with respect to variations of the path we must somehow restrict the set of paths we are con- sidering to contain only one realizable path. It will turn out that for Lagrangians that depend only on the configuration and rate of change of configuration it is enough to restrict the set of paths to those that have the same configuration at the endpoints of the path segment. The Principle of Stationary Action14 asserts that for each dy- namical system we can cook up a Lagrangian such that a realizable path connecting the configurations at two times t1 and t2 is dis- tinguished from all conceivable paths by the fact that the action S[γ](t1, t2) is stationary with respect to variations of the path. For Lagrangians that depend only on the configuration and rate of change of configuration the variations are restricted to those that preserve the configurations at t1 and t2.15 14 The principle is often called the “Principle of Least Action” because its initial formulations spoke in terms of the action being minimized rather than the more general case of taking on a stationary value. The term “Principle of Least Action” is also commonly used to refer to a result, due to Maupertuis, Euler, and Lagrange, which says that free particles move along paths for which the integral of the kinetic energy is minimized among all paths with the given endpoints. Correspondingly, the term “action” is sometimes used to refer specifically to the integral of the kinetic energy. (Actually, Euler and Lagrange used the vis viva, or twice the kinetic energy.) 15 Other ways of stating the principle of stationary action make it sound teleo- logical and mysterious. For instance, one could imagine that the system con- siders all possible paths from its initial configuration to its final configuration and then chooses the one with the smallest action. Indeed, the underlying vi- sion of a purposeful, economical, and rational universe played no small part in the philosophical considerations that accompanied the initial development of
- 35. 1.2 Configuration Spaces 9 Exercise 1.1: Fermat optics Fermat observed that the laws of reflection and refraction could be ac- counted for by the following facts: Light travels in a straight line in any particular medium with a velocity that depends upon the medium. The path taken by a ray from a source to a destination through any sequence of media is a path of least total time, compared to neighboring paths. Show that these facts do imply the laws of reflection and refraction.16 1.2 Configuration Spaces Let us consider mechanical systems that can be thought of as composed of constituent point particles, with mass and position, but with no internal structure.17 Extended bodies may be thought of as composed of a large number of these constituent particles with specific spatial relationships between them. Extended bodies maintain their shape because of spatial constraints between the constituent particles. Specifying the position of all the constituent particles of a system specifies the configuration of the system. The existence of constraints between parts of the system, such as those that determine the shape of an extended body, means that the constituent particles cannot assume all possible positions. The set of all configurations of the system that can be assumed is called the configuration space of the system. The dimension of the mechanics. The earliest action principle that remains part of modern physics is Fermat’s Principle, which states that the path traveled by a light ray between two points is the path that takes the least amount of time. Fermat formu- lated this principle around 1660 and used it to derive the laws of reflection and refraction. Motivated by this, the French mathematician and astronomer Pierre-Louis Moreau de Maupertuis enunciated the Principle of Least Action as a grand unifying principle in physics. In his Essai de cosmologie (1750) Maupertuis appealed to this principle of “economy in nature” as evidence of the existence of God, asserting that it demonstrated “God’s intention to regu- late physical phenomena by a general principle of the highest perfection.” For a historical perspective of Maupertuis’s, Euler’s, and Lagrange’s roles in the formulation of the principle of least action, see Jourdain [25]. 16 For reflection the angle of incidence is equal to the angle of reflection. Re- fraction is described by Snell’s law. Snell’s Law is that when light passes from one medium to another, the ratio of the sines of the angles made to the normal to the interface is the inverse of the ratio of the refractive indices of the media. The refractive index is the ratio of the speed of light in the vacuum to the speed of light in the medium. 17 We often refer to a point particle with mass but no internal structure as a point mass.
- 36. 10 Chapter 1 Lagrangian Mechanics configuration space is the smallest number of parameters that have to be given to completely specify a configuration. The dimension of the configuration space is also called the number of degrees of freedom of the system.18 For a single unconstrained particle it takes three parameters to specify the configuration. Thus the configuration space of a point particle is three dimensional. If we are dealing with a system with more than one point particle, the configuration space is more com- plicated. If there are k separate particles we need 3k parameters to describe the possible configurations. If there are constraints among the parts of a system the configuration is restricted to a lower-dimensional space. For example, a system consisting of two point particles constrained to move in three dimensions so that the distance between the particles remains fixed has a five-dimensional configuration space: for example, with three numbers we can fix the position of one particle, and with two others we can give the position of the other particle relative to the first. Consider a juggling pin. The configuration of the pin is specified if we give the positions of every atom making up the pin. However, there exist more economical descriptions of the configuration. In the idealization that the juggling pin is truly rigid, the distances among all the atoms of the pin remain constant. So we can specify the configuration of the pin by giving the position of a single atom and the orientation of the pin. Using the constraints, the positions of all the other constituents of the pin can be determined from this information. The dimension of the configuration space of the juggling pin is six: the minimum number of parameters that specify the position in space is three, and the minimum number of parameters that specify an orientation is also three. As a system evolves with time, the constituent particles move subject to the constraints. The motion of each constituent particle 18 Strictly speaking the dimension of the configuration space and the number of degrees of freedom are not the same. The number of degrees of freedom is the dimension of the space of configurations that are “locally accessible.” For systems with integrable constraints the two are the same. For systems with non-integrable constraints the configuration dimension can be larger than the number of degrees of freedom. For further explanation see the discussion of systems with non-integrable constraints below (section 1.10.3). Apart from that discussion, all of the systems we will consider have integrable constraints (they are “holonomic”). This is why we have chosen to blur the distinction be- tween the number of degrees of freedom and the dimension of the configuration space.
- 37. 1.3 Generalized Coordinates 11 is specified by describing the changing configuration. Thus, the motion of the system may be described as evolving along a path in configuration space. The configuration path may be specified by a function, the configuration-path function, which gives the configuration of the system at any time. Exercise 1.2: Degrees of freedom For each of the mechanical systems described below, give the number of degrees of freedom of the configuration space. a. Three juggling pins. b. A spherical pendulum, consisting of a point mass hanging from a rigid massless rod attached to a fixed support point. The pendulum bob may move in any direction subject to the constraint imposed by the rigid rod. The point mass is subject to the uniform force of gravity. c. A spherical double pendulum, consisting of one point-mass hanging from a rigid massless rod attached to a second point-mass hanging from a second massless rod attached to a fixed support point. The point mass is subject to the uniform force of gravity. d. A point mass sliding without friction on a rigid curved wire. e. A top consisting of a rigid axisymmetric body with one point on the symmetry axis of the body attached to a fixed support, subject to a uniform gravitational force. f. The same as e, but not axisymmetric. 1.3 Generalized Coordinates In order to be able to talk about specific configurations we need to have a set of parameters that label the configurations. The param- eters that are used to specify the configuration of the system are called the generalized coordinates. Consider an unconstrained free particle. The configuration of the particle is specified by giving its position. This requires three parameters. The unconstrained particle has three degrees of freedom. One way to specify the po- sition of a particle is to specify its rectangular coordinates relative to some chosen coordinate axes. The rectangular components of the position are generalized coordinates for an unconstrained par- ticle. Or consider an ideal planar double pendulum: a point mass constrained to always be a given distance from a fixed point by a rigid rod, with a second mass that is constrained to be at a given distance from the first mass by another rigid rod, all confined to a
- 38. 12 Chapter 1 Lagrangian Mechanics vertical plane. The configuration is specified if the orientation of the two rods is given. This requires at least two parameters; the planar double pendulum has two degrees of freedom. One way to specify the orientation of each rod is to specify the angle it makes with the vertical. These two angles are generalized coordinates for the planar double pendulum. The number of coordinates need not be the same as the dimen- sion of the configuration space, though there must be at least that many. We may choose to work with more parameters than neces- sary, but then the parameters will be subject to constraints that restrict the system to possible configurations, that is, to elements of the configuration space. For the planar double pendulum described above, the two angle coordinates are enough to specify the configuration. We could also take as generalized coordinates the rectangular coordinates of each of the masses in the plane, relative to some chosen coordinate axes. These are also fine coordinates, but we will have to explicitly keep in mind the constraints that limit the possible configurations to the actual geometry of the system. Sets of coordinates with the same dimension as the configuration space are easier to work with because we do not have to deal with explicit constraints among the coordinates. So for the time being we will consider only formulations where the number of configuration coordinates is equal to the number of degrees of freedom; later we will learn how to handle systems with redundant coordinates and explicit constraints. In general, the configurations form a space M of some dimen- sion n. The n-dimensional configuration space can be parametrized by choosing a coordinate function χ that maps elements of the configuration space to n-tuples of real numbers. If there is more than one dimension, the function χ is a tuple of n independent coordinate functions19 χi, i = 0, . . . , n − 1, where each χi is a real-valued function defined on some region of the configuration space.20 For a given configuration m in the configuration space M 19 A tuple of functions that all have the same domain is itself a function on that domain: Given a point in the domain the value of the tuple of functions is a tuple of the values of the component functions at that point. 20 The use of superscripts to index the coordinate components is traditional, even though there is potential confusion, say, with exponents. We use zero- based indexing.
- 39. 1.3 Generalized Coordinates 13 the values χi(m) of the coordinate functions are the generalized coordinates of the configuration. These generalized coordinates permit us to identify points of the n-dimensional configuration space with n-tuples of real numbers.21 For any given configura- tion space, there are a great variety of ways to choose generalized coordinates. Even for a single point moving without constraints, we can choose rectangular coordinates, polar coordinates, or any other coordinate system that strikes our fancy. The motion of the system can be described by a configuration path γ mapping time to configuration-space points. Correspond- ing to the configuration path is a coordinate path q = χ◦γ mapping time to tuples of generalized coordinates. If there is more than one degree of freedom the coordinate path is a structured object: q is a tuple of component coordinate path functions qi = χi ◦ γ. At each instant of time t, the values q(t) = (q0(t), . . . , qn−1(t)) are the generalized coordinates of a configuration. The derivative Dq of the coordinate path q is a function22 that gives the rate of change of the configuration coordinates at a given time: Dq(t) = (Dq0(t), . . . , Dqn−1(t)). The rate of change of a generalized coordinate is called a generalized velocity. We can make coordinate representations for higher derivatives of the path as well. We introduce the function (pronounced 21 More precisely, the generalized coordinates identify open subsets of the con- figuration space with open subsets of Rn . It may require more than one set of generalized coordinates to cover the entire configuration space. For example, if the configuration space is a two-dimensional sphere, we could have one set of coordinates that maps (a little more than) the northern hemisphere to a disk, and another set that maps (a little more than) the southern hemisphere to a disk, with a strip near the equator common to both coordinate systems. A space that can be locally parametrized by smooth coordinate functions is called a differentiable manifold. The theory of differentiable manifolds can be used to formulate a coordinate-free treatment of variational mechanics. An introduction to mechanics from this perspective can be found in [2] or [5] . 22 The derivative of a function f is a function. It is denoted Df. Our notational convention is that D is a high-precedence operator. Thus D operates on the adjacent function before any other application occurs: Df(x) is the same as (Df)(x).
- 40. 14 Chapter 1 Lagrangian Mechanics “chart”) that extends a coordinate representation to the local tu- ple:23 χ(t, γ(t), Dγ(t), . . .) = (t, q(t), Dq(t), . . .) , (1.5) where q = χ ◦ γ. The function χ takes the coordinate-free local tuple (t, γ(t), Dγ(t), . . .) and gives a coordinate representation as a tuple of the time, the value of the coordinate path function at that time, and the values of as many derivatives of the coordinate path function as are needed. Given a coordinate path q = χ◦γ the rest of the local tuple can be computed from it. We introduce a function Γ that does this Γ[q](t) = (t, q(t), Dq(t), . . .) . (1.6) The evaluation of Γ only involves taking derivatives of the coordi- nate path q = χ ◦ γ; the function Γ does not depend on χ. From relations (1.5) and (1.6) we find Γ[q] = χ ◦ T [γ]. (1.7) Exercise 1.3: Generalized coordinates For each of the systems described in exercise 1.2 specify a system of generalized coordinates that can be used to describe the behavior of the system. Lagrangians in generalized coordinates The action is a property of a configuration path segment for a particular Lagrangian L. The action does not depend on the co- ordinate system that is used to label the configurations. We can use this property to find a coordinate representation Lχ for the Lagrangian L. 23 The formal definition of is unimportant to the discussion, but if you really want to know here is one way to do it: First, we define the derivative Dγ of a configuration path γ in terms of ordinary derivatives by specifying how it acts on sufficiently smooth real- valued functions f of configurations: (Dn γ)(t)(f) = Dn (f ◦ γ)(t). Then we define χ(a, b, c, d, . . .) = (a, χ(b), c(χ), d(χ), . . .) . With this definition: χ(t, γ(t), Dγ(t), D2 γ(t), . . .) = ¡ t, χ(γ(t)), Dγ(t)(χ), D2 γ(t)(χ), . . . ¢ = ¡ t, χ ◦ γ(t), D(χ ◦ γ)(t), D2 (χ ◦ γ)(t), . . . ¢ = ¡ t, q(t), Dq(t), D2 q(t), . . . ¢ .
- 41. 1.3 Generalized Coordinates 15 The action is S[γ](t1, t2) = Z t2 t1 L ◦ T [γ]. (1.8) The Lagrangian L is a function of the local tuple T [γ](t) = (t, γ(t), Dγ(t), . . .). The local tuple has the coordinate represen- tation Γ[q] = χ ◦ T [γ], where q = χ ◦ γ. So if we choose24 Lχ = L ◦ −1 χ , (1.9) then25 Lχ ◦ Γ[q] = L ◦ T [γ]. (1.10) On the left we have the composition of functions that use the intermediary of a coordinate representation; on the right we have the composition of two functions that do not involve coordinates. We define the coordinate representation of the action to be Sχ[q](t1, t2) = Z t2 t1 Lχ ◦ Γ[q]. (1.11) The function Sχ takes a coordinate path; the function S takes a configuration path. Since the integrands are the same by equa- tion (1.10) the integrals have the same value: S[γ](t1, t2) = Sχ[χ ◦ γ](t1, t2). (1.12) So we have a way of constructing coordinate representations of a Lagrangian that gives the same action for a path in any coordinate system. For Lagrangians that depend only on positions and velocities the action can also be written Sχ[q](t1, t2) = Z t2 t1 Lχ (t, q(t), Dq(t)) dt. (1.13) 24 The coordinate function χ is locally invertible, and so is χ. 25 L ◦ T [γ] = L ◦ −1 χ ◦ χ ◦ T [γ] = Lχ ◦ Γ[χ ◦ γ] = Lχ ◦ Γ[q].
- 42. 16 Chapter 1 Lagrangian Mechanics The coordinate system used in the definition of a Lagrangian or an action is usually unambiguous, so the subscript χ will usually be dropped. 1.4 Computing Actions To illustrate the above ideas, and to introduce their formulation as computer programs, we consider the simplest mechanical system— a free particle moving in three dimensions. Euler and Lagrange discovered that for a free particle the time-integral of the kinetic energy over the particle’s actual path is smaller than the same integral along any alternative path between the same points: a free particle moves according to the principle of stationary action, provided we take the Lagrangian to be the kinetic energy. The ki- netic energy for a particle of mass m and velocity ~ v is 1 2mv2, where v is the magnitude of ~ v. In this case we can choose the generalized coordinates to be the ordinary rectangular coordinates. Following Euler and Lagrange, the Lagrangian for the free par- ticle is26 L(t, x, v) = 1 2m(v · v), (1.14) where the formal parameter x names a tuple of components of the position with respect to a given rectangular coordinate sys- tem, and where the formal parameter v names a tuple of velocity components.27 We can express this formula as a procedure: 26 Here we are making a function definition. A definition specifies the value of the function for arbitrarily chosen formal parameters. One may change the name of a formal parameter, so long as the new name does not conflict with any other symbol in the definition. For example, the following definition specifies exactly the same free-particle Lagrangian: L(a, b, c) = 1 2 m(c · c). 27 The Lagrangian is formally a function of the local tuple, but any particular Lagrangian only depends on a finite initial segment of the local tuple. We define functions of local tuples by explicitly declaring names for the elements of the initial segment of the local tuple that includes the elements upon which the function depends.
- 43. 1.4 Computing Actions 17 (define ((L-free-particle mass) local) (let ((v (velocity local))) (* 1/2 mass (dot-product v v)))) The definition indicates that L-free-particle is a procedure that takes mass as an argument and returns a procedure that takes a local tuple local,28 extracts the generalized velocity with the procedure velocity, and uses the velocity to compute the value of the Lagrangian. Suppose we let q denote a coordinate path function that maps time to position components:29 q(t) = (x(t), y(t), z(t)) . (1.15) We can make this definition30 (define q (up (literal-function ’x) (literal-function ’y) (literal-function ’z))) where literal-function makes a procedure that represents a function of one argument that has no known properties other than the given symbolic name.31 The symbol q now names a procedure 28 We represent the local tuple as a composite data structure, the components of which are the time, the generalized coordinates, the generalized velocities, and possibly higher derivatives. We do not want to be bothered by the details of packing and unpacking the components into these structures, so we provide utilities for doing this. The constructor ->local takes the time, the coor- dinates, and the velocities and returns a data structure representing a local tuple. The selectors time, coordinate, and velocity extract the appropri- ate pieces from the local structure. The procedures time = (component 0), coordinate = (component 1) and velocity = (component 2). 29 Be careful. The x in the definition of q is not the same as the x that was used as a formal parameter in the definition of the free-particle Lagrangian above. There are only so many letters in the alphabet, so we are forced to reuse them. We will be careful to indicate where symbols are given new meanings. 30 A tuple of coordinate or velocity components is made with the procedure up. Component i of the tuple q is (ref q i). All indexing is zero based. The word up is to remind us that in mathematical notation these components are indexed by superscripts. There are also down tuples of components that are indexed by subscripts. See the appendix on notation. 31 In our system, arithmetic operators are generic over symbols and expressions as well as numeric values; so arithmetic procedures can work uniformly with numbers or expressions. For example, if we have the procedure (define (cube
- 44. 18 Chapter 1 Lagrangian Mechanics of one real argument (time) that produces a tuple of three com- ponents representing the coordinates at that time. For example, we can evaluate this procedure for a symbolic time t as follows: (print-expression (q ’t)) (up (x t) (y t) (z t)) The procedure print-expression produces a printable form of the expression. The procedure print-expression simplifies ex- pressions before printing them. The derivative of the coordinate path Dq is the function that maps time to velocity components: Dq(t) = (Dx(t), Dy(t), Dz(t)). We can make and use the derivative of a function.32 For example, we can write: (print-expression ((D q) ’t)) (up ((D x) t) ((D y) t) ((D z) t)) The function Γ takes a coordinate path and returns a function of time that gives the local tuple (t, q(t), Dq(t), . . .). We implement this Γ with the procedure Gamma. Here is what Gamma does: (print-expression ((Gamma q) ’t)) (up t (up (x t) (y t) (z t)) (up ((D x) t) ((D y) t) ((D z) t))) So the composition L ◦ Γ is a function of time that returns the value of the Lagrangian for this point on the path: (print-expression ((compose (L-free-particle ’m) (Gamma q)) ’t)) (+ (* 1/2 m (expt ((D x) t) 2)) (* 1/2 m (expt ((D y) t) 2)) (* 1/2 m (expt ((D z) t) 2))) x) (* x x x)) we can obtain its value for a number (cube 2) => 8 or for a literal symbol (cube ’a) => (* a a a). 32 Derivatives of functions yield functions. For example, ((D cube) 2) => 12 and ((D cube) ’a) => (* 3 (expt a 2)).
- 45. 1.4 Computing Actions 19 The procedure show-expression is like print-expression except that it puts the simplified expression into traditional infix form and displays the result.33 Most of the time we will use this method of display, to make the boxed expressions that appear in this book. It also produces the prefix form as returned by print-expression, but we will usually not show this.34 (show-expression ((compose (L-free-particle ’m) (Gamma q)) ’t)) 1 2 m (Dx (t))2 + 1 2 m (Dy (t))2 + 1 2 m (Dz (t))2 According to equation (1.11) we can compute the Lagrangian action from time t1 to time t2 as: (define (Lagrangian-action L q t1 t2) (definite-integral (compose L (Gamma q)) t1 t2)) Lagrangian-action takes as arguments a procedure L that com- putes the Lagrangian, a procedure q that computes a coordinate path, and starting and ending times t1 and t2. The definite- integral used here takes as arguments a function and two lim- its t1 and t2, and computes the definite integral of the function over the interval from t1 to t2.35 Notice that the definition of Lagrangian-action does not depend on any particular set of co- ordinates or even the dimension of the configuration space. The method of computing the action from the coordinate representa- tion of a Lagrangian and a coordinate path does not depend on the coordinate system. We can now compute the action for the free particle along a path. For example, consider a particle moving at uniform speed 33 The display is generated with TEX. 34 For very complicated expressions the prefix notation of Scheme is often bet- ter, but simplification is almost always useful. We can separate the functions of simplification and infix display. We will see examples of this later. 35 Scmutils includes a variety of numerical integration procedures. The ex- amples in this section were computed by rational-function extrapolation of Euler-MacLaurin formulas with a relative error tolerance of 10−10 .
- 46. 20 Chapter 1 Lagrangian Mechanics along a straight line t 7→ (4t + 7, 3t + 5, 2t + 1).36 We represent the path as a procedure (define (test-path t) (up (+ (* 4 t) 7) (+ (* 3 t) 5) (+ (* 2 t) 1))) For a particle of mass 3, we obtain the action between t = 0 and t = 10 as37 (Lagrangian-action (L-free-particle 3.0) test-path 0.0 10.0) 435. Exercise 1.4: Lagrangian actions For a free particle an appropriate Lagrangian is38 L(t, x, v) = 1 2 mv2 . Suppose that x is the constant-velocity straight-line path of a free par- ticle, such that xa = x(ta) and xb = x(tb). Show that the action on the solution path is m 2 (xb − xa)2 tb − ta . Paths of minimum action We already know that the actual path of a free particle is uniform motion in a straight line. According to Euler and Lagrange the action is smaller along a straight-line test path than along nearby paths. Let q be a straight-line test path with action S[q](t1, t2). Let q + ²η be a nearby path, obtained from q by adding a path 36 Surely for a real physical situation we would have to specify units for these quantities. In this illustration we do not give units. 37 Here we use decimal numerals to specify the parameters. This forces the representations to be floating point, which is efficient for numerical calculation. If symbolic algebra is to be done it is essential that the numbers be exact integers or rational fractions, so that expressions can be reliably reduced to lowest terms. Such numbers are specified without a decimal point. 38 The squared magnitude of the velocity is ~ v · ~ v, the vector dot-product of the velocity with itself. The square of a structure of components is defined to be the sum of the squares of the individual components, so we write simply v2 = v · v.
- 47. 1.4 Computing Actions 21 variation η scaled by the real parameter ².39 The action on the varied path is S[q + ²η](t1, t2). Euler and Lagrange found S[q + ²η](t1, t2) > S[q](t1, t2) for any η that is zero at the endpoints and for any small non-zero ². Let’s check this numerically by varying the test path, adding some amount of a test function that is zero at the endpoints t = t1 and t = t2. To make a function η that is zero at the endpoints, given a sufficiently well-behaved function ν, we can use η(t) = (t − t1)(t − t2)ν(t). This can be implemented: (define ((make-eta nu t1 t2) t) (* (- t t1) (- t t2) (nu t))) We can use this to compute the action for a free particle over a path varied from the given path, as a function of ²:40 (define ((varied-free-particle-action mass q nu t1 t2) epsilon) (let ((eta (make-eta nu t1 t2))) (Lagrangian-action (L-free-particle mass) (+ q (* epsilon eta)) t1 t2))) The action for the varied path, with ν(t) = (sin t, cos t, t2), and ² = 0.001 is, as expected, larger than for the test path: ((varied-free-particle-action 3.0 test-path (up sin cos square) 0.0 10.0) 0.001) 436.29121428571153 39 Note that we are doing arithmetic on functions. We extend the arithmetic operations so that the combination of two functions of the same type (same domains and ranges) is the function on the same domain that combines the values of the argument functions in the range. For example, if f and g are functions of t, then fg is the function t 7→ f(t)g(t). A constant multiple of a function is the function whose value is the constant times the value of the function for each argument: cf is the function t 7→ cf(t). 40 Note that we are adding procedures. Paralleling our extension of arithmetic operations to functions, arithmetic operations are extended to compatible pro- cedures.
- 48. 22 Chapter 1 Lagrangian Mechanics We can numerically compute the value of ² for which the action is minimized. We search between, say −2 and 1:41 (minimize (varied-free-particle-action 3.0 test-path (up sin cos square) 0.0 10.0) -2.0 1.0) (-1.5987211554602254e-14 435.0000000000237 5) We find exactly what is expected—that the best value for ² is zero,42 and the minimum value of the action is the action along the straight path. Finding trajectories that minimize the action We have used the variational principle to determine if a given trajectory is realizable. We can also use the variational princi- ple to actually find trajectories. Given a set of trajectories that are specified by a finite number of parameters, we can search the parameter space looking for the trajectory in the set that best ap- proximates the real trajectory by finding one that minimizes the action. By choosing a good set of approximating functions we can get arbitrarily close to the real trajectory.43 One way to make a parametric path that has fixed endpoints is to use a polynomial that goes through the endpoints as well as a number of intermediate points. Variation of the positions of the intermediate points varies the path; the parameters of the varied path are the coordinates of the intermediate positions. The procedure make-path constructs such a path using a Lagrange 41 The arguments to minimize are a procedure implementing the univariate function in question, and the lower and upper bounds of the region to be searched. Scmutils includes a choice of methods for numerical minimization; the one used here is Brent’s algorithm, with an error tolerance of 10−5 . The value returned by minimize is a list of 3 numbers: the first is the argument at which the minimum occurred, the second is the minimum obtained, and the third is the number of iterations of the minimization algorithm required to obtain the minimum. 42 Yes, -1.5987211554602254e-14 is zero for the tolerance required of the min- imizer. And the 435.0000000000237 is arguably the same as 435 obtained before. 43 There are lots of good ways to make such a parametric set of approximating trajectories. One could use splines or higher-order interpolating polynomials; one could use Chebyshev polynomials; one could use Fourier components. The choice depends upon the kinds of trajectories one wants to approximate.
- 49. 1.4 Computing Actions 23 interpolation polynomial.44 The procedure make-path is called with five arguments: (make-path t0 q0 t1 q1 qs), where q0 and q1 are the endpoints, t0 and t1 are the corresponding times, and qs is a list of intermediate points. Having specified a parametric path we can construct a paramet- ric action that is just the action computed along the parametric path: (define ((parametric-path-action Lagrangian t0 q0 t1 q1) qs) (let ((path (make-path t0 q0 t1 q1 qs))) (Lagrangian-action Lagrangian path t0 t1)))) We can find approximate solution paths by finding parameters that minimize the action. We do this minimization with a canned multidimensional minimization procedure:45 (define (find-path Lagrangian t0 q0 t1 q1 n) (let ((initial-qs (linear-interpolants q0 q1 n))) (let ((minimizing-qs (multidimensional-minimize (parametric-path-action Lagrangian t0 q0 t1 q1) initial-qs))) (make-path t0 q0 t1 q1 minimizing-qs)))) 44 Here is one way to implement make-path: (define (make-path t0 q0 t1 q1 qs) (let ((n (length qs))) (let ((ts (linear-interpolants t0 t1 n))) (Lagrange-interpolation-function (append (list q0) qs (list q1)) (append (list t0) ts (list t1)))))) The procedure linear-interpolants produces a list of elements that linearly interpolate the first two arguments. We use this procedure here to specify ts, the n evenly spaced intermediate times between t0 and t1 at which the path will be specified. The parameters being adjusted, qs, are the positions at these intermediate times. The procedure Lagrange-interpolation-function takes a list of values and a list of times and produces a procedure that computes the Lagrange interpolation polynomial that goes through these points. 45 The minimizer used here is the Nelder-Mead downhill simplex method. As usual with numerical procedures, the interface to the nelder-mead procedure is complex, with lots of optional parameters to allow the user to control errors effectively. For this presentation we have specialized nelder-mead by wrapping it in the more palatable multidimensional-minimize. Unfortunately, you will have to learn to live with complicated numerical procedures someday.
- 50. 24 Chapter 1 Lagrangian Mechanics The procedure multidimensional-minimize takes a procedure (in this case the value of the call to action-on-parametric-path) that computes the function to be minimized (in this case the action) and an initial guess for the parameters. Here we choose the initial guess to be equally-spaced points on a straight line between the two endpoints, computed with linear-interpolants. To illustrate the use of this strategy, we will find trajectories of the harmonic oscillator, with Lagrangian46 L(t, q, v) = 1 2 mv2 − 1 2kq2, (1.16) for mass m and spring constant k. This Lagrangian is imple- mented by (define ((L-harmonic m k) local) (let ((q (coordinate local)) (v (velocity local))) (- (* 1/2 m (square v)) (* 1/2 k (square q))))) We can find an approximate path taken by the harmonic oscil- lator for m = 1 and k = 1 between q(0) = 1 and q(π/2) = 0 as follows:47 (define q (find-path (L-Harmonic 1.0 1.0) 0. 1. :pi/2 0. 3)) We know that the trajectories of this harmonic oscillator, for m = 1 and k = 1, are q(t) = A cos(t + ϕ) (1.17) where the amplitude A and the phase ϕ are determined by the initial conditions. For the chosen endpoint conditions the solution is q(t) = cos(t). The approximate path should be an approxima- tion to cosine over the range from 0 to π/2. Figure 1.1 shows the error in the polynomial approximation produced by this process. The maximum error in the approximation with three intermedi- ate points is less than 1.7 × 10−4. We find, as expected, that the error in the approximation decreases as the number of intermedi- 46 Don’t worry. We know that you don’t yet know why this is the right La- grangian. We will get to this in section 1.6. 47 By convention, named constants have names that begin with colon. The constants named :pi and :-pi are what we would expect from their names.
- 51. 1.4 Computing Actions 25 π/2 π/4 0 +0.0002 0 -0.0002 Figure 1.1 The difference between the polynomial approximation with minimum action and the actual trajectory taken by the harmonic oscillator. The abscissa is the time and the ordinate is the error. ate points is increased. For four intermediate points it is about a factor of 15 better. Exercise 1.5: Solution process We can watch the progress of the minimization by modifying the proce- dure parametric-path-action to plot the path each time the action is computed. Try this: (define win2 (frame 0. :pi/2 0. 1.2)) (define ((parametric-path-action Lagrangian t0 q0 t1 q1) intermediate-qs) (let ((path (make-path t0 q0 t1 q1 intermediate-qs))) ;; display path (graphics-clear win2) (plot-function win2 path t0 t1 (/ (- t1 t0) 100)) ;; compute action (Lagrangian-action Lagrangian path t0 t1))) (find-path (L-harmonic 1. 1.) 0. 1. :pi/2 0. 2) Exercise 1.6: Minimizing action Suppose we try to obtain a path by minimizing an action for an im- possible problem. For example, suppose we have a free particle and we
- 52. 26 Chapter 1 Lagrangian Mechanics impose endpoint conditions on the velocities as well as the positions that are inconsistent with the particle being free. Does the formalism protect itself from such an unpleasant attack? You may find it illuminating to program it and see what happens. 1.5 The Euler-Lagrange Equations The principle of stationary action characterizes the realizable paths of systems in configuration space as those for which the action has a stationary value. In elementary calculus, we learn that the critical points of a function are the points where the derivative vanishes. In an analogous way, the paths along which the action is stationary are solutions of a system of differential equations. This system, called the Euler-Lagrange equations or just the Lagrange equations, is the link that permits us to use the principle of stationary action to compute the motions of me- chanical systems, and to relate the variational and Newtonian formulations of mechanics.48 Lagrange equations We will find that if L is a Lagrangian for a system that depends on time, coordinates, and velocities, and if q is a coordinate path for which the action S[q](t1, t2) is stationary (with respect to any variation in the path that keeps the endpoints of the path fixed) then D(∂2L ◦ Γ[q]) − ∂1L ◦ Γ[q] = 0. (1.18) Here L is a real-valued function of a local tuple; ∂1L and ∂2L denote the partial derivatives of L with respect to its general- ized position and generalized velocity arguments.49 The function ∂2L maps a local tuple to a structure whose components are the derivatives of L with respect to each component of the gener- alized velocity. The function Γ[q] maps time to the local tuple: Γ[q](t) = (t, q(t), Dq(t), . . .). Thus the compositions ∂1L◦Γ[q] and 48 This result was initially discovered by Euler and later rederived by Lagrange. 49 The derivative or partial derivative of a function that takes structured argu- ments is a new function that takes the same number and type of arguments. The range of this new function is itself a structure with the same number of components as the argument with respect to which the function is differenti- ated.
- 53. 1.5.1 Derivation of the Lagrange Equations 27 ∂2L◦Γ[q] are functions of one argument, time. The Lagrange equa- tions assert that the derivative of ∂2L ◦ Γ[q] is equal to ∂1L ◦ Γ[q], at any time. Given a Lagrangian, the Lagrange equations form a system of ordinary differential equations that must be satisfied by realizable paths.50 1.5.1 Derivation of the Lagrange Equations We will show that Principle of Stationary Action implies that realizable paths satisfy a set of ordinary differential equations. First we will develop tools for investigating how path-dependent functions vary as the paths are varied. We will then apply these tools to the action, to derive the Lagrange equations. Varying a path Suppose that we have a function f[q] that depends on a path q. How does the function vary as the path is varied? Let q be a coordinate path and q + ²η be a varied path, where the function η is a path-like function that can be added to the path q, and the factor ² is a scale factor. We define the variation δηf[q] of the function f on the path q by51 δηf[q] = lim ²→0 µ f[q + ²η] − f[q] ² ¶ . (1.19) 50 Lagrange’s equations are traditionally written in the form d dt ∂L ∂q̇ − ∂L ∂q = 0, or, if we write a separate equation for each component of q, as d dt ∂L ∂q̇i − ∂L ∂qi = 0 i = 0, . . . , n − 1 . In this way of writing Lagrange’s equations the notation does not distinguish between L, which is a real-valued function of three variables (t, q, q̇), and L ◦ Γ[q], which is a real-valued function of one real variable t. If we do not realize this notational pun, the equations don’t make sense as written—∂L/∂q̇ is a function of three variables, so we must regard the arguments q, q̇ as functions of t before taking d/dt of the expression. Similarly, ∂L/∂q is a function of three variables, which we must view as a function of t before setting it equal to d/dt(∂L/∂q̇). These implicit applications of the chain rule pose no problem in performing hand computations—once you understand what the equations represent. 51 The variation operator δη is like the derivative operator in that it acts on the immediately following function: δηf[q] = (δηf)[q].
- 54. 28 Chapter 1 Lagrangian Mechanics The variation of f is a linear approximation to the change in the function f for small variations in the path. The variation of f depends on η. A simple example is the variation of the identity path function: I[q] = q. Applying the definition δηI[q] = lim ²→0 µ (q + ²η) − q ² ¶ = η. (1.20) It is traditional to write δηI[q] simply as δq. Another example is the variation of the path function that returns the derivative of the path. We have δηg[q] = lim ²→0 µ D(q + ²η) − Dq ² ¶ = Dη with g[q] = Dq. (1.21) It is traditional to write δηg[q] as δDq. The variation may be represented in terms of a derivative. Let g(²) = f[q + ²η], then δηf[q] = lim ²→0 µ g(²) − g(0) ² ¶ = Dg(0). (1.22) Variations have the following derivative-like properties. For path-dependent functions f and g and constant c: δη(f g)[q] = δηf[q] g[q] + f[q] δηg[q] (1.23) δη(f + g)[q] = δηf[q] + δηg[q] (1.24) δη(cf)[q] = c δηf[q]. (1.25) Let F be a path-independent function and let g be a path-dependent function, then δηh[q] = (DF ◦ g[q]) δηg[q] with h[q] = F ◦ g[q]. (1.26) The operators D (differentiation) and δ (variation) commute in the following sense: Dδηf[q] = δηg[q] with g[q] = D(f[q]). (1.27) Variations also commute with integration in a similar sense. If a path-dependent function f is stationary for a particular path q with respect to small changes in that path then it must be
- 55. Exploring the Variety of Random Documents with Different Content
- 56. land enclosed bore to the whole area of the county, but the proportion which it bore to the whole area available for cultivation. This, which is of course not ascertainable, is clearly a very different thing.[462] It is no consolation to a family which has been evicted from a prosperous farm to be told that it can settle on a moor or a marsh, on Blackstone Edge or Deeping Fen. To argue that enclosing was of little consequence, because so small a proportion of the total land area was enclosed, is almost precisely similar to arguing that overcrowding is of little consequence, because the area of Great Britain divided by the population gives a quotient of about one and a half acres to every human being in the country. The evidence of a general trend of opinion during a century and a half—opinion by no means confined to the peasants, or to the peasants' champions like Hales, or to idealists like Sir Thomas More, or to the preachers of social righteousness like Latimer and Crowley, but shared by Wolsey and Thomas Cromwell in the earlier part of the century, Robert Cecil and Francis Bacon[463] at the end of it—to the effect that the agrarian changes caused extensive depopulation, is really a firmer basis for judging their effects than are statistics which, however carefully worked up, are necessarily unreliable, and which, when reliable, are not quite the statistics required. When that opinion is backed by documentary proof that from one village thirty persons, from another fifty, from another the whole population, were displaced, though of course we cannot say that such displacement was general, we can say that it was not unknown, and that if contemporaries were guilty of exaggeration (as they probably were), their exaggeration took the form not of inventing extreme cases, but of suggesting that such extreme cases were the rule. On the whole, therefore, our conclusions as to the quantitative measurement of depopulation caused in the sixteenth century must still, in spite of the researches of Mr. Leadam and Professor Gay, be a negative one. In the first place, we cannot say, even approximately, what proportion of the total landholding population was displaced. In the second place, such figures as we do possess are not of a kind to outweigh the direct evidence of contemporary observers that the
- 57. movement was so extensive as in parts of England to cause serious suffering and disturbance. (d) The Agrarian Changes and the Poor LawToC The obscurity in which the statistics of depopulation are involved does not prevent us from seeing that it played an important part in providing an incentive to the organisation of relief on a national and secular basis, which was the most enduring achievement of the social legislation of sixteenth century statesmen. An influential theory of Poor Law History regards the admission finally made in 1601 that the destitute person has, not only a moral, but a legal, right to maintenance, as a last fatal legacy handed to the modern state by the expiring social order of the Middle Ages, a relic of villeinage which was given a statutory basis at the very moment when a little more patience would have shown that a national system of poor relief was not only unnecessary, but positively harmful, in the new mobile society which the expansion of commerce and industry was bringing into existence. “Serfdom,” says an eminent exponent of this view, “is itself a system of Poor Law. The Poor Law is not therefore a new device invented in the time of Elizabeth to meet a new disease. The very conception of a society based on status involves the conception of a Poor Law far more searching and rigid than the celebrated 43 Eng. cap. 2.... The collective provision is appropriate to the then expiring condition of status.... A wide diffusion of private property, not collective property, is the obvious and natural method by which the unable-bodied periods of life are to be met. With the disappearance of Feudalism we might have expected that there would have disappeared the custom which made the poor a charge upon the manor or parish of which they had formerly been serfs. This, however, did not happen, and a history of this survival of mediæval custom is the history of the English Poor Law.... To sum the matter up:—In following the development of Poor Law legislation, we watch society struggling to free itself from the fetters of a primitive communism of poverty and subjection, a state of
- 58. things possessing many 'plausible advantages.' Legislation for the management of the Poor often impeded, and only occasionally expedited, this beneficent process.... It proceeded from ignorance of the true nature of progress, and from a denial or neglect of the power of absorption possessed by a free society.”[464] It is obvious that in this passage Mr. Mackay uses his interpretation of Poor Law origins to make a very trenchant criticism upon the whole principle involved in the public maintenance of the destitute. That principle was not introduced because new conditions made its adoption indispensable. It survived from an older order of things into a world in which the only serious causes of destitution are personal and not economic, and in which therefore it is quite inappropriate. To tolerate it is to drag for ever a clanking chain, one end of which is fastened round the bleeding ankles of modern society, and the other anchored in the hideous provisions of the Statute of Labourers. Nor should we be wrong if we said that a similar theory, though less lucidly expressed, has had a considerable influence upon Poor Law practice. For the idea of a Poor Law as an anachronism which is quite out of place in a developed economic society is implied more than once in the celebrated report drafted by Senior and Chadwick in 1834, and has passed from that brilliant piece of special pleading into the minds of three generations of administrators. “A person,” they state, “who attributes pauperism to the inability to procure employment, will doubt the efficiency of the cause which we propose to remove it," whereas “whenever inquiries have been made as to the previous condition of the able-bodied individuals who live in such numbers on the town parishes, it has been found that the pauperism of the greater number has originated in indolence, improvidence, and vice, and might have been avoided by ordinary care and industry. The majority of the Statutes connected with the administration of public relief have created new evils, and aggravated those which they were intended to prevent.”[465] A discussion of Poor Law theory and history falls outside the limits of this essay. But in forming an estimate of the effects of the agrarian changes which have been described above, it is perhaps not out of
- 59. place to consider the minor question of the connection between them and the system of Poor Relief which took its final shape in the reign of Elizabeth. Since the distress which the relief institutions of an age exist to meet stands to its general economic conditions in the relation of reverse to obverse, of effect to cause, of disease to environment, much light is thrown on the economic difficulties most characteristic of any period by ascertaining the type of distress with which relieving authorities are most generally confronted. Equally important, any student of Poor Law History, who is not the partisan of a theory, finds himself constantly driven to look for an explanation of Poor Law developments in regions which, at first sight, appear to lie far outside his immediate subject, but where, in reality, is grown the grim harvest which it is the duty of Poor Law authorities, often acting in complete ignorance of its origin, to reap. Much wild theorising and some tragic practical blunders might have been avoided, had it been more generally realised that, of all branches of administration, the treatment of persons in distress is that which can least bear to be left to the exclusive attention of Poor Law specialists, because it, most of all matters, depends for its success on being carefully adapted to the changing economic conditions, the organisation or disorganisation of industry, the stability or instability of trade, the diffusion or concentration of property, by which the nature and extent of the distress requiring treatment are determined. When one turns to the age in which the Poor Law took shape, the first thing to strike one is that the need for it arises, according to the views expressed by most writers of the period, from that very development in commercial relationships, that very increase in economic mobility, which Mr. Mackay seems to imply should have made it unnecessary. The special feature of sixteenth century pauperism is written large over all the documents of the period—in Statutes, in Privy Council proceedings, in the records of Quarter Sessions. The new and terrible problem is the increase in vagrancy. The sixteenth century lives in terror of the tramp. He is denounced by moralists, analysed into species by the curious or scientific,
- 60. scourged and buffeted by all men. The destitution of the aged and impotent, of fatherless children and widows, is familiar enough. It has been with the world from time immemorial. It has been for centuries the object of voluntary charitable effort; and when the dissolution of the monasteries dries up one great channel of provision, the Government intervenes with special arrangements[466] to take their place a whole generation before it can be brought to admit that there is any problem of the unemployed, other than the problem of the sturdy rogue. The distinction between the able- bodied unemployed and the impotent is one which is visible to the eye of sense. The distinction between the man who is unemployed because he cannot get work and the man who is unemployed because he does not want work, requires a modicum of knowledge and reflection which even at the present day is not always forthcoming. The former distinction, therefore, is not supplemented by the latter until the beginning of the last quarter of the century. [467] In one respect, that of the Law of Settlement, the English Poor Law does show traces of a mediæval origin. In all other respects, so far from being a survival from the Middle Ages, it comes into existence just at the time when mediæval economic conditions are disappearing. It is not accepted at once as a matter of course that the destitute shall be publicly relieved, still less that the able-bodied destitute deserve anything but punishment. Governments make desperate efforts for about one hundred years to evade their new obligations. They whip and brand and bore ears; they offer the vagrant as a slave to the man who seizes him; they appeal to charity; they introduce the parish clergy to put pressure on the uncharitable; they direct the bishops to reason with those who stop their ears against the parish clergy. When merely repressive measures and voluntary effort are finally discredited, they levy a compulsory charge rather as a fine for contumacy than as a rate, and slide reluctantly into obligatory assessments[468] only when all else has failed. And if we ask why the obligation of maintaining the destitute should have received national recognition first in the sixteenth century, we can only answer by pointing to that trend away from the stationary conditions of agriculture to the fluctuating
- 61. conditions of trade, and in particular to that displacement of the rural population, which we have already seen was one result of enclosure. The national Poor Law is not a mediæval anachronism. It is the outcome of conditions which seem to the men of the sixteenth century new and appalling. Of these conditions the most important are the agrarian changes. Let us try for a moment to put ourselves in the position of a family which has been evicted from its holding to make room for sheep. When the last stick of furniture has been tumbled out by the bailiff, where, poor houseless wretches, are they to turn? They cannot get work in their old home, even if they can get lodgings, for the attraction of sheep-farming is that the wage bill is so low. Will they emigrate from England like the Scotch crofters? There are people who in the seventeenth century will advise them to seek a haven with the godly folk who have crossed the Atlantic, who will argue that England is overstocked, that “there is such pressing and oppressing in town and country about farms, trades, traffic, so as a man can hardly anywhere set up a trade but he shall pull down two of his neighbours,” and point out that “the country is replenished with new farmers, and the almhouses are filled with old labourers,” that “the rent-taker lives on sweet morsels, but the rent-payer eats a dry crust often with watery eyes.”[469] But enclosures have been going on for a century before the plantations exist to offer a refuge, and in any case the probability of the country folk hearing of them is very remote. Can a man migrate to seek work in another part of the country? Not easily, for, apart from the enormous practical difficulties, the law puts obstacles in his way, and the law is backed up with enthusiasm by every parish and town in the country. There are three possible attitudes which a State may adopt towards the questions arising from the ebb and flow of population. It may argue, with the optimists of 1834, that the mobility of labour is a good thing, a symptom of alertness and energy, and that it will take place of itself to the extent which is economically desirable, provided that no impediments are placed in the way of those who desire to better themselves by looking for work elsewhere. Or, while believing that it
- 62. is much to be desired that people should migrate freely from place to place in search of employment, it may nevertheless reflect that the mere absence of restrictions does not in fact stimulate such movement, and therefore take upon itself its encouragement through the publication of information and the registration of unemployed workers. Or, subordinating economic to political considerations, it may hold that the movement of a large number of unemployed persons up and down the country is not an indication of a praiseworthy spirit of enterprise, but a menace to public order which must be sternly repressed. We need hardly say that this last view is the one characteristic of the sixteenth century. The attitude towards the man on tramp in search of employment is exactly the opposite of that which is held at the present day. He is not less, but much more, culpable than he who remains in his own parish and lives on his neighbours. He is assumed not to be seeking work but to be avoiding it, and avoiding it in a restless and disorderly manner. Hear what the worthy Harrison says when the State has already made the provision for the unemployed a charge upon each parish:—“But if they refuse to be supported by this benefit of the law, and will rather endeavour by going to and fro to maintain their idle trades, then are they adjudged to be parcel of the third sort (i.e. wilful vagrants), and so, instead of courteous refreshing at home, are often corrected with sharp execution and whip of justice abroad. Many there are which, notwithstanding the rigour of the laws provided on that behalf, yield rather with this liberty (as they call it) to be daily under the fear and terror of the whip, than by abiding where they were born or bred, to be provided for by the devotion of the parishes.”[470] The village is still thought of as the unit of employment. It is still regarded as being equipped with the means of finding work for all its inhabitants, as though there had been no movement towards pasture-farming to prick a hole in its economic self-sufficiency. The presumption, therefore, is against the man who leaves the parish where he is known to his neighbours. He must prove that he is going to take up work for which he is already engaged. He must get a licence from his last employer. As far as the able-bodied are concerned the Poor Law is in origin a measure of social police. Relief is thrown in as a
- 63. makeweight, because by the end of the sixteenth century our statesmen have discovered that when economic pressure reaches a certain point they cannot control men without it. The whip has no terrors for the man who must look for work or starve. So every Sunday after church, while Parson’s sermon is still fresh in our minds, we board out our poor by rotation “among such householders as will maintain them meat and work and such wages as they shall deserve for the week following.”[471] Heaven help us if the next parish does not do the same! And the Poor Law is a police measure for the necessity of which the agrarian changes are largely responsible. In spite of all the obstacles in the way of migration, in spite of whip and courteous refreshment, men do in fact migrate, and not only men, but women and children. By the latter part of the century, at any rate, statesmen have begun to understand that pauperism and vagrancy stand to the depopulation caused by enclosure in the relation of effect to cause. The revolution in the official attitude to the problem caused by this belated illumination is as great as that which has taken place in the last ten years with regard to unemployment. Once the new standpoint has been seized, though opinion, and the opinion not only of the ruling classes, but of burgesses and villagers, still treats the vagrant with iron severity, it never quite relapses into the comfortable doctrine, the grand discovery of a commercial age, that distress is itself a proof of the demerits of its victim, and that Heaven, like a Utilitarian philosopher, permits the existence of destitution only that it may make “less eligible" the lot of “improvidence and vice.” It is saved from this last error not by the lore of economists, but because it regards economic questions through the eyes of a sturdy matter-of-fact morality. It is sufficiently enlightened to recognise that even among vagrants there is a class which is more sinned against than sinning, a class of whom it can be asked “at whose hands shall the blood of these men be required?”[472] It is sufficiently ingenuous to answer by pointing to “some covetous man" who, “espying a further commodity in their commons, holds, and tenures, doth find such means as thereby to
- 64. wipe many out of their occupyings and turn the same unto his private gains.”[473] Occasionally the effect of enclosures is brought home to the encloser in a practical way, by compelling him not only to pay a fine to the Crown, but also to make a contribution towards the relief of the poor whose numbers he has increased.[474] To see the way in which the relation between the problems of pauperism and of agrarian depopulation is regarded, turn to the debates in the House of Commons. In the year 1597, when both questions are acute (the preceding year had seen a recrudescence of agrarian rioting), a member or minister, probably Robert Cecil, is preparing notes for a speech[475] on the subject in Parliament. What are the points he emphasises? They are the high price of corn caused by bad harvests and the manipulations of middlemen, the enclosing of land and the conversion of arable to pasture, which naturally intensifies the difficulty of securing adequate food supplies, “the decaying and plucking down of houses, ... and not only the plucking down of some few houses, but the depopulating of whole towns ... and keeping of a shepherd only, whereby many subjects are turned without habitation, and fill the country with rogues and idle persons.” When Parliament meets in October, the House is at once busy with different aspects of the same question.[476] Bills are introduced dealing with forestallers, regrators, and engrossers of corn, with vagrancy and pauperism, and with enclosures, and a committee is appointed to consider the latter question. In the debates which follow there is the usual division of opinion between the champions of economic reform and the advocates of more, and more ruthless, “deterrence," between those who wish to legislate as to causes and those who are mainly occupied with symptoms. Bacon, master as ever of the science of his subject, insists with invincible logic that pauperism is one part of the general agrarian problem, and he is supported by Robert Cecil. On the other hand, the experts as to pauperism—we can imagine the county justices fresh from their whippings and relief committees and houses of correction, fresh, too, from enclosure and depopulation—complain that their special subject is being overlooked in a general and
- 65. dangerous discussion on the economic causes of distress, and that the committee “has spent all their travel about the said enclosures and tillage, and nothing about the said rogues and poor.” That this should have been the popular line to take needs no explanation. A Parliament which dares discuss not only how to manipulate the lives of the poor, but the fundamental causes of their misery, is a Parliament which the eye of man had not yet, has not yet, beheld. Compared with other representative assemblies, compared with itself at a later date, the Elizabethan House of Commons, debating in an age when it could be said that government was “nothing but a certein conspiracy of riche men procuringe theire owne commodities under the name and title of the Common Wealth,” had the grace to show some stirrings of compunction. If members who had grown fat on the tragedy which they were discussing spoke of their victims as members will speak, ministers at least were independent, and could venture, like Cecil, to tell the House unpalatable truths. Of the two Acts against enclosure, which were the result of this session's deliberations, we shall speak later. What is worth noticing here is the disposition, even in a Parliament composed of country gentlemen, to emphasise the connection between the problems with which anti- enclosure and anti-vagrancy legislation have to deal. It is summed up in the eloquent peroration of a nameless member. “As this bill entered at first with a short prayer, 'God speed the plough,' so I wish it may end with such success as the plough shall speed the poor.”[477] What became of the families displaced from the soil between their final eviction and that subsidence upon the stony breast of the Elizabethan Poor Law, which, for some of them, was their ultimate fate? There is no certain information to guide us. The tragedy of the tramp is his isolation. Every man’s hand is against him; and his history is inevitably written by his enemies. Yet, beneath denunciations hurled upon him by those who lived warm and slept soft, we can see two movements going on, two waves in a vast and silent ebbing of population from its accustomed seats. In the first place there is a steady immigration into the towns on the part of
- 66. those “who, being driven out of their habitations, are forced into the great cities, where, being very burdensome, men shut their doors against them, suffering them to die in the streets and highways.”[478] The municipal records of the periods teem with complaints of the disorder, the overcrowding, the violation of professional bye-laws, caused by rural immigration. The displaced peasant is the Irishman of the sixteenth century, and, like the Irishman, he makes his very misery a whip with which to scourge, not alas! his oppressors, but men who often are not much less wretched than himself. He turns whole quarters into slums, spreads disease through congested town dwellings, and disorganises the labour market by crowding out the native artisan. Gild members find themselves eaten up by unlawful men who have never served an apprenticeship in the town, and retort with regulations requiring the deposit of a prohibitive sum as an entrance fee from all immigrants who want to set up shop, especially from those wretches who are thought to have a large family of children, at present snugly concealed in their last place of residence, but soon to be surreptitiously introduced, a brood of hungry young cuckoos, if once their parents get a footing in the town.[479] Borough authorities, who see cottages “made down" into tenements in which pestilence spreads with fearful rapidity, seek to stamp out the very possibility of invasion by prohibiting the erection of new cottages or the subdivision of old. To judge by their behaviour, the notorious Statute of 1662, which codified the existing customs as to settlement, must have been one of the most popular pieces of legislation ever passed by Parliament. Town[480] after town in the course of the sixteenth century tries to protect itself by a system of stringent inspection worthy of modern Germany. Sometimes there is a regular expulsion of the aliens. “Forasmuch as it is found by daily experience,” declare the authorities of Nottingham,[481] “that by the continual building and erecting of new cottages and poor habitations, and by the transferring of barns and suchlike buildings into cottages, and also by the great confluence of many poor people from forrein parts out of this towne to inhabit here, and lykewise by the usual and frequent taking in of inmates into many poor habitations here, the poorer sort
- 67. of people do much increase ... it is ordered that no burgess or freeman on pain of £5 erect any cottage or convert any building into a cottage in the town without license of the Mayor, that no burgess or freeman, without a license, receive any one from the country as a tenant, that every landlord be bound in the sum of £10 to remove all foreign tenants who have entered in the last three years before May 1st next.” What most boroughs do for themselves is finally, after many regulations have been made by the Common Council, done for London by Parliamentary legislation. It is not a chance that the end of Elizabeth’s reign sees the first two Housing Acts, one[482] in 1589, enacting that only one family may live in a house, the other[483] applying to London alone, and forbidding the division of houses into tenements, the receiving of lodgers, or the erection of new houses for persons who are assessed in the subsidy book at less than £5 in goods or £3 in lands. The evicted peasants are beginning to take their revenge. They have been taking it ever since. In the second place there is a general movement from the enclosed to the open field villages. The families displaced by enclosure cannot easily enter into industry, even if they wish to do so, for the avenue to most trades is blocked both by the Corporations and by the statutory system of a seven years' apprenticeship, which maintains professional standards at the expense of an unprivileged residuum. What they do is to follow the orthodox advice given to those who have lost their customary means of livelihood. They proceed to colonise, and to colonise in such numbers that they cannot easily be kept out. They settle as squatters on the waste lands of those manors which have not been enclosed, and which, before the waste is turned into a sheep-run, offer no obstacle to immigration. That the possibility of using the manorial waste to accommodate those who had no settled abode had occurred to statesmen as one expedient for meeting the problem of the infirm and destitute, is shown by the sanction expressly given in the Poor Law of 1597[484] to the expenditure of parish funds on the erection of cottages on the waste as residences for the impotent poor. In fact, however, the mobility of labour was becoming such that it was impossible, even if it had been
- 68. desirable, to reserve those unutilised territories for the maintenance of the impotent. In spite of bitter protests from the existing inhabitants, refugees from other villages swarm down upon them in such numbers that the Act requiring four acres of land to be attached to each cottage cannot be observed, and the issuing of licences for the erection of cottages on the waste for able-bodied men, who have come with their families from a distance, becomes a regular part of the business of Quarter Sessions.[485] Such a redistribution of the population solves one problem only to create others. Stern economists in the seventeenth century lament that the ease with which permission to build cottages on the waste is obtained encourages the existence of an improvident and idle class, which will neither work for wages nor make good use of the land. “In all or most towns where the fields lie open and are used in common, there is a new brood of upstart intruders as inmates, and the inhabitants of unlawful cottages erected contrary unto law.... Loyterers who will not usually be got to work unless they may have such excessive wages as they themselves desire.”[486] The opponents of enclosure answer with some justice that, in effect, the open field villages are saddled with the destitution caused by enclosing landlords, who first ruin their tenants and then, like a modern Dock Company which relies on the Poor Rate to save its wage-bill, leave them to be supported by those places to which they are compelled to migrate.[487] The latter difficulty is indeed a very serious one, which not only is the occasion of numberless petitions[488] from villages who wish to be assisted by, or to avoid assisting, their neighbours, but on occasion converts even the country gentry into opponents of enclosure. “We further conceive,” write the Justices of Nottingham to the Council, “that if depopulation may be reformed it will bring a great good to the whole Kingdom; for where homes are pulled down the people are forced to seek new habitations in other towns and countries, whereof those towns where they get a settling are pestered so as they are hardly able to live one by another, and it is likewise the cause of erecting new cottages upon the waste and other places who are not able to relieve themselves ... which causes rogues and vagabonds to
- 69. increase.”[489] In the elaborate book of Poor Law orders published in 1631 the Government recognises the genuineness of this grievance, and, to its direction that richer parishes should contribute funds to the aid of the poor, adds a special rider pointing out that such extra contributions would come with special appropriateness from those places where there had been depopulation. We may now summarise our view of the social effects of the changes introduced by lords of manors, and by the capitalist farmers who manage their estates. When the demesne land is enclosed and converted to pasture, there is an appreciable diminution in the demand for labour, and consequently an increase in unemployment. When the common rights of tenants are curtailed, they lose not only an important subsidiary source of income, but often, at the same time, the means of cultivating their arable holdings. When their holdings are merged in the great estate of the capitalist farmer, they are turned adrift to seek their living in a world where most trades and most towns are barred against them, where they are punished if they do not find work, and punished if they look for work without permission, where “if the poor being thrust out of their houses go to dwell with others, straight we catch them with the Statute of Inmates; if they wander abroad, they are in danger of the Statute of the Poor to be whipped.”[490] Thus, quite apart both from the eternal source of poverty which consists in the recalcitrance of nature to human effort, and from those causes of individual destitution which in all ages and in all economic conditions lie in wait for the exceptionally unfortunate or the exceptionally improvident, for the sick, the aged, and the orphan, there is an increase in the number of those for whom access to the land, their customary means of livelihood, is unobtainable, and consequently a multiplication of the residuum for whom the haunting insecurity of the propertyless modern labourer is, not the exception, but the normal lot. It is this extension of destitution among able-bodied men, who have the will, but not the means, to find employment, which is the peculiar feature of sixteenth century pauperism, and which leads in 1576 to the most characteristic expedient of the Elizabethan Poor Law—the provision
- 70. of materials upon which the unemployed can be set to work. The recognition that the relief of the destitute must be enforced as a public obligation was not the consequence of the survival of mediæval ideas into an age where they were out of place, but an attempt on the part of the powerful Tudor state to prevent the social disorder caused by economic changes, which, in spite of its efforts, it had not been strong enough to control.[Next Chapter] FOOTNOTES: [416] The Shepe Book of Tittleshall Manor (Holkham MSS., Tittleshall Books, No. 19), shows flocks of 500 to 1000 sheep being managed by a single shepherd, 1543–1549. [417] e.g. Holkham MSS., Fulmordeston, Bdle. 6: “To the Right Honourable Sir Edward Cooke, Knight, Attorney General unto the King’s Matie. Humblie sheweth unto your lordship yor poore and dayley orators ... yor worshippes tenants of the Manor of Fulmordeston cum Croxton in the Duchie of Lancaster, and the moste parte of the tenants of the same manor that whereas your said orators in the Hillary Terme last commenced suite in the Duchie Courte against Thomas Odbert and Roger Salisbury, gent., who have enclosed their grounds contrary to the custom of the manor, wherby your wor. loseth your shack due out of the grounds, common lane or way for passengers is stopped up, and your worshipps' poore orators lose their accustomed shack in those grounds, and the said Roger Salisbury taketh also the whole benefit of theire common from them, keepinge there his sheepe in grazinge, and debarring them of their libertie there which for comon right belongeth unto them.” For the rest of this document see Appendix I., and compare the following defence to a charge of breaking open an enclosure: “The owners of the said tenements, from time whereof there is no memory to the contrary, have had a common of pasture for themselves and their tenants in one close commonly called 'the new leasue,' in the lordship of Weston in the manner following; that is to say, when the field where the said 'leasue' doth lie, called Radnor field, lieth fallow, then through the whole year; and when the said field is sown with corn, then from the reaping and carrying away of the corn until the same be sown again ... and the said Thomas Dodd further said that he did break
- 71. open the said close ... being fenced in such time as he ought to have common in the same, to the end that his cattle might take their pasture therein" (William Salt Collection, New Series, vol. ix., Chancery Proceedings, Bdle. 8, No. 9). [418] For complaints of tenants against the exactions, of farmers as early as 1413, see Victoria County History, Essex, vol. ii. p. 318. For a stipulation in the farmer's covenant, see the following: “Item a covenant conteyned in this lease that the said Thomas shall permit and suffer the customary Tenants peaceably to have and enjoy their estates, rights, grants, interests, and premises, without any lette, interruption, or contradiction of the said Thomas" (Roxburghe Club, Pembroke Surveys, Knyghton); and Northumberland County History, vol. v. p. 208, Buston: “The tenants of this town at the beginning of summer have their oxen allway grazed in Shilbottel wood, or else they were not able to maintain their tenements. It is therefore requisite that his lordship or his heire should have respect unto the want of pasture, that in any lease made by his lordship or his heire to any person of the pasture, the said Shilbottel wood, there might be a proviso in the said lease that the said tenants should have their oxen ground there, as they have been accustomed.” Instances of the harrying of the peasants by the large farmers are to be found, ibid., vol. i. p. 350 (Tughall), and p. 274 (Newham). [419] All Souls' Archives, vol. i. p. 203, No. 356. [420] Topographer and Genealogist, vol. i., Survey of Mudford and Hinton. In this case the aggressor was not the farmer of the demesne, but a freeholder owning a third of the manor. To escape his depredations the tenants proposed “to enclose their common fieldes and to assign to Master Lyte and his tenants his third parte in every field by itself, and to extinguish his right of common in the rest.” [421] Victoria County History, Suffolk, “Social and Economic History.” [422] For an amusing example see Conway, The Alps from End to End, pp. 190–192. [423] The Commonweal of this Realm of England, p. 57. [424] Ten acres of “turf” to forty acres of arable was the estimate of his requirements made to me by an Oxfordshire small holder.
- 72. [425] Topographer and Genealogist, vol. i.: “The tenants of Landress have common in a certayne ground called King’s Moore for all kinde of cattle, and every one of them may keep in the said moore as much of all kind of cattle in somer as their severall or ingrounde will beare in the wynter, whyche is a great relief to the poore tenants, for as they confesse they keep all their cattle there in the somer, and reserve their ingroundes untouched for the winter.” [426] e.g. Southampton Court Leet Records (Hearnshaw), pp. 4– 5, 1550: “Item we present that no burgers or comyners at one time comyn above the number of two beasts upon payne of every such defaulte 2s.; provided that iff any of them have two kyne or wenlings, he shall have no horse, and yf he have but one cow he may have one horse.” [427] Topographer and Genealogist, vol. i.—Rolleston (Stafford): “The said manor is ... well inhabited with divers honest men, whose trade of lyvinge is onlie by husbandry ... and have no large pastures or severall closes ... but have been alwaie accustomed to have their cattle and sometyme their ploughe beasts pastured in the Queen's Majestie’s Park of Rolleston, for xxd., the stage ... without which aide and help they were neither able to maintain hospitallitie nor tyllage; and nowe of late yeares the fermor of the herbage hath advanced the stage to 6s. 8d., and yet the Quene’s Majesties rent nothing increased.” [428] Fitzherbert, Book of Husbandry. [429] Northumberland County History, vol. v., Birling: “Allowed part of 25s. 4d. for focage of Orchard Medow and Mylneside Bank, because they are now enclosed within the lord’s new Park, and this allowance shall be made yearly until the tenants of Byrling have and peacefully enjoy another parcel of pasture to the same value 11s. 8d." (Bailiff’s Accounts, 1474). R.O. Misc. Books Land Rev., vol. ccxx., f. 236: “Divers parcels of land and pasture of the manor of Farfield, now common of 140 acres, now occupied by the tenants there as commons and given them in exchange in satisfaction of their old common imparked in the new Park, £6, 13s. 8d.” [430] Pollock and Maitland, History of English Law, vol. i. p. 606. For the questions concerning common rights see ibid., pp. 594– 624, and Maitland, Domesday Book and Beyond, pp. 340–356; Vinogradoff, Villainage in England, Essay II. chap, ii., and The
- 73. Growth of the Manor, Book II. chap. iv. I have followed Vinogradoff’s rather than Maitland’s view. [431] For buying and selling of pasture see below, and for enclosure pp. 168–170. The following seems a clear case of more or less corporate action. Holkham MSS., Burnham, Bdle. 5, No. 94: “Copy of an indenture between [here follows a list of names] of the same town and county, yeomen, as well on the behalf of themselves as of the rest of the comoners and freeholders of the said town of the one part, and Robert Bacon of [illegible] in the County of Norfolk, and Thomas Coke of Grays Inn in the County of Middlesex of the other part, that whereas heretofore Sir Philip [illegible] being lord and owner of the marshes hereafter mentioned ... did by his indenture of bargain and sale bearing date ... 1588, grant bargain and sell unto [list of names as above] all those marsh grounds lying and being in Burnham, to have and to hold the said premises to the parties last before mentioned and their heires to the use of them and their heires for ever, to the intent and purpose notwithstanding that the said parties last before mentioned there, being inhabitants in certain ancient messuages in the said Towne, and all other inhabitants of the said Towne there and afterwards for the tyme being in any of the ancient messuages and cottages in the said towne, for so long time as they shall be there inhabitinge and noe longer, according to the quantity of their tenures within the said Towne might depasture and feede the land as by the said deeds referring thereunto being had may more fully appeare; [it recites that the land] may by wallinge and embankinge the same be improved to more than a [illegible] value, and made fitt for arrable, meadowe, and pasture grounde, whereby tillage may be increased and his Majestie’s subjects receive more employment thereby, and danger of drawing [drowning?] of their stock for their feedinge prevented [recites that Robert Bacon and Thomas Coke have undertaken to drain the land in return for receiving three parts of it and that the persons above mentioned] being the major parte of the parties interested in the said salte Marshes, and being enabled by the lawes and Statutes of this realm to contract and bargaine with any person or persons for the draining thereof" [now convey 3 parts of the marshes to the above-mentioned Robert Bacon and Thomas Coke], June 8, 1637. The motive of this agreement was to get the low-lying meadows on the sea-coast drained. Drainage schemes were much in the air about this time, and any one who has seen the country near Holkham and Burnham will know how
- 74. badly protection from the sea was needed. Two points are worth noticing: (i.) the tenants have no objection to surrendering part of their common if they get a quid pro quo; (ii.) they act as a single body. They buy land and they sell land and they can leave it to their heirs. Certain persons in the township act on their behalf, much as directors might act for a body of shareholders. Is it possible to speak of such arrangements simply in terms of individual rights? Are we not driven to think of the township as almost a landholding corporation? [432] Common appendant, common appurtenant, common in gross, and common par cause de vicinage. This classification is not found in Bracton, and appears to date from the late Middle Ages, see Vinogradoff, Villainage in England, Essay II., chap, ii., and the following case: Coke’s Reports, Part IV., p. 60. Hill, 4 Jac. I. in Communi Banco: “Robert Smith brought an action of Trespass against Stephen Gatewood, gent., quare clausum fregit ... cum quibusdam averiis.... Defendant pleaded a certain custom, 'quod inhabitantes infra eandem villam de Stixwood prædictam infra aliquod antiquum messuagium ibidem ratione commorantiæ et residentiæ suæ in eadem habuerunt et usi fuerunt et consueverunt habere com. Pastur ... pro omnibus et omnimodis bobus et equis et aliis grossis animalibus.' Unanimously resolved that the custom is against law. 1. That there are but four manners of common, common appendant, appurtenant, in gross, and by reason of vicinage, and this common ratione commorantiæ is none of them. 2. What estate shall he have, who is inhabitant, in the common, when it appears he hath no estate or interest in the house (but a mere habitation and dwelling) in respect of which he ought to have his common? For none can have interest in a common in respect of a house in which he hath no interest.” [433] Coke, Complete Copyholder, Sect. 53: “When an Act of Parliament altereth the service, tenure, or interest of the land, or other thing in prejudice of the lord or of the Customs of the Manor, or in prejudice of the tenant, then the generall words of such an Act of Parliament extend not to the copyhold; but when an Act is generally made for the good of the commonwealth, and no prejudice may accrue by reason of the alteration of any interest, service, tenure, or Custom, of the Manor, there usually copyhold lands are within the generall purview of such Acts.” [434] Fitzherbert, Book of Surveying: “And as for that manner of common, me seemeth the Lord may improve himself of their
- 75. waste grounds, leaving their own tenants sufficient common, having no regard to the tenants of the other lordship. But as far as all errable lands, meadows, leises, and pastures, the lordes may improve themselves by course of the common law, for the statute speaketh nothing but of waste grounds.” [435] e.g. Coventry Leet Book, vol. ii. p. 510. [436] Genealoger and Archæologist, vol. i., Manor of West Coker (Somerset): “The demesnes remayneth in one entier ferm, and is dymysed to one Sir John Seymour, knight, who being confederate with the freeholders of the manor, maketh such inclosers for his owne lucre, and suffreth the freeholders to do the same, nevertheless surcharge the common with their cattle, that in process of tyme yt wilbe the destruccion of the custumarye tenants.” [437] For a discussion of the legal position of the copyholders see below, pp. 287–310. [438] Coventry Leet Book, vol. ii. pp. 445–446 and passim. [439] If the common was so large that it had been unnecessary to “stint” it, why did the city object to the lord putting additional beasts on? I take the situation to be that the Prior—probably tempted by the profitableness of sheep-farming in the latter part of the fifteenth century—diminished the pasture which the city could use, by putting on many more beasts than ever before, which, in the absence of a recognised “stint,” he was able to do without violating any custom, as he would have done if there had been a customary limit, as on many manors. [440] Topographer and Genealogist, vol. iii. These are the people whom Heaven protected in the way described on p. 148 note. Observe what this little community endured. (i.) Sir Francis Englefield, senior, seizes 1900 out of 2000 acres of their common. (ii.) Sir Francis Englefield, junior, seizes “the charter of our town ... and the deed of the said common." (iii.) He tries to seize the remaining 100 acres, and ruins them by lawsuits “for the space of seven or eight years at the least, and never suffers any one to come to triall in all that space ... that the said Free tenants were not able to wage law any longer, for one John Rous ... was thereby enforced to sell all his land (to the value of £500) with following the suits in law, and many were thereby impoverished." (iv.) He turns them out of their shops in the market-place, and
- 76. introduces instead “a stranger that liveth not in the town." (v.) He appoints his own nominee as mayor, in defiance of the custom which requires him to appoint one of two men submitted to him by the jury. (vi.) He prevents his victims from signing this petition by threats of eviction. ("They are fearful that they shall be put forth of their bargaines, and then they shall not tell how to live, otherwise they would have set to their hands.") [441] Holkham MSS., Map of West Lexham. [442] R.O. Aug. Off. Misc. Bks., vol. cccxcix., f. 201 ff. [443] The manors are South Newton, Winterbourne Basset, Knyghton, Donnington, and Estoverton and Phipheld (Roxburghe Club, Surveys of Pembroke Manors). [444] This, of course, is not inconsistent with a general appreciation, i.e. a general rise in wages and fall in the rate of interest. [445] Northumberland County History, vol. ix. p. 124. For a similar case of evictions by Delavale, showing how they were carried out, ibid., pp. 201–202: “There was in Seaton Delavale township 12 tenements, whereon there dwelt 12 able men sufficiently furnished with horse and furniture to serve his Majestie ... who paid 46s. 8d. rent yearlie a piece or thereabouts. All the said tenants and their successors saving 5 the said Robert Delavale eyther thrust out of their fermholds or weried them by taking excessive fines, increasing of their rents unto £3 a piece, and withdrawing part of their best land and meadow from their tenements ... by taking their good land from them and compelling them to winne moorishe and heathe ground, and after their hedging heth ground to their great charge, and paying a great fine, and bestowing great reparation on building their tenements, he quite thrust them off in one yeare, refusing either to repay the fine or to repay the charge bestowed in diking or building.... The said seven fermholds displaced had to every one of them 60 acres of arable land, viz. 20 in every field at the least, as the tenants affirme, which amounteth to 480 acres of land yearlie or thereabouts, converted for the most part from tillage to pasture, and united to the demaine of the lordship of Seaton Delavale.” [446] In several cases the freeholders' lands are not stated in the survey, and are therefore not included in this table.
- 77. [447] A few acres described as “held without title" are omitted. [448] I am not sure that there are not other lands in Domerham not included in the survey or in the demesne. If this is so, the proportion of the latter to the rest of the manorial land would of course be reduced. [449] R.O. Rentals and Surveys, Gen. Ser., Portf. 22, No. 18. [450] Roxburghe Club, Surveys of Pembroke Manors. [451] Ibid., and Hoare, History of Wiltshire, Hundred of Ambresbury. [452] Northumberland County History, vol. i. p. 350. [453] Ibid., vol. ix., Cowpen. [454] Ibid., vol. i. p. 275. [455] Ibid., vol. ix. pp. 201–202. [456] Moore, The Crying Sin of England, &c. [457] Cal. S. P. D. Eliz., 1595–1597 (p. 347), quoted Gay, Quarterly Journal of Economics, vol. xvii. [458] “Certayne Causes gathered together wherein is shewed the decaye of England only by the great multitude of shepe" (E. E. T. S. date 1550–1553). “It is to understande ... that there is in England townes and villages to the number of fifty thousand and upward, and for every town and village ... there is one plough decayed since the fyrst year of the reign of King Henry VII.... The whiche 50,000 ploughs every plough was able to maintain 6 persons, and nowe they have nothing, but goeth about in England from dore to dore.” [459] For a discussion of the value of these reports see Leadam, Domesday of Enclosures, and Trans. Royal Hist. Soc., New Series, vol. vi.; Gay, Trans. Royal Hist. Soc., New Series, vol. xiv. and vol. xviii.; Gay, Quarterly Journal of Economics, vol. xvii. (1902–1903). A useful summary of the evidence, with a map illustrating the probable geographical distribution of the movement, is given by Johnson, The Disappearance of the Small Landowner, pp. 42–54 and Map I. [460] It is a question how far there had ever been an open field system in some of these counties, e.g. Cornwall and Kent. There
- 78. certainly were some open field villages of the ordinary pattern in Kent (see Slater, The English Peasantry and the Enclosure of Common Fields, p. 230). But Kent from an early date develops on its own lines, and does not go through the same stages of manorialism and commutation as other counties. Much of it seems to start at the point which they reach only in the sixteenth century. Cornwall again, though in the sixteenth century there were commons where the villagers pastured their cattle together (see accounts of Landress and Porpehan, Topographer and Genealogist, vol. i.), was largely a county of scattered homesteads and very early enclosure (for the “nucleated village" and “scattered homesteads,” see Maitland, Domesday Book and Beyond, pp. 15–16), pointing to a different system of settlement from that of the counties where the open field system obtained. For enclosures in Devon and Somerset see Cunningham, Growth of English Industry and Commerce, Modern Times, Part II., App. B: “A consideration of the cause in question before the lords touchinge depopulation," and Carlyle’s Cromwell, Letter XXIV. “Lest we should engage our body of horse too far into that enclosed country.” [461] For intimidation see the case of Wootton Basset, quoted above, pp. 251–253, and below, pp. 302–304. Also Gay, Trans. Royal Hist. Soc., New Series, vol. xviii.; and Hales' defence (appendix to Miss Lamond’s introduction to The Commonweal of this Realm of England). [462] Professor Pollard has good remarks on this point (Political History of England, 1547–1603, p. 29). [463] Wolsey was responsible for the Commission of 1517. For a letter of Cromwell to Henry VIII. on the subject of enclosure, and for the views of Cecil and Bacon, see below, pp. 273–274, 279, 343, 387. [464] Mackay, History of the English Poor Law, 1834–1898, pp. 10–11, 16–17. [465] Poor Law Commission Report of 1834, pp. 264–277, 281. [466] 27 Hen. VIII., c. 25. Under this Act city and county authorities are to relieve impotent beggars “by way of voluntary and charitable alms.” They are also for the first time given power to apprentice vagrant children.
- 79. [467] 18 Eliz. c. 3 directed that a stock of wool, flax, hemp, iron, or other stuff should be provided in cities, corporate towns, and market towns. The important words which show the change of opinion are, “To the intente also that ... Roges ... may not have any just excuse in saying they cannot get any service or work.” [468] 14 Eliz. c. 5. [469] Robert Cushman, “Reasons and Considerations touching the Lawfulness of Removing out of England into the parts of America" (printed by E. Arber, The Story of the Pilgrim Fathers). [470] Harrison in Elizabethan England (Withington), chap. x. [471] Hist. MSS. Com., Marquis of Salisbury, Part VII., pp. 160– 161: “Orders agreed to by the Justices of the Peace for Cornwall at General Sessions for Bodmin the 5th and Truro the 8th of April, 39 Eliz.” [472] Harrison, loc. cit. [473] Ibid. [474] Camden Society, 1886. Cases in Courts of Star Chamber and High Commission, Michaelmas, 7 Caroli, Case of Archer. (The allusion in the text is to a precedent cited in this case.) [475] Hist. MSS. Com., Marquis of Salisbury, Part VII., Nov. 1597. “Notes for the present Parliament.” [476] D'Ewes' Journal, pp. 551–555; see also Leonard, The Early History of English Poor Relief, pp. 73–75. [477] Hist. MSS. Com., Marquis of Salisbury, Part VII., pp. 541– 543. [478] Lansd. MSS. 83, f. 68, quoted Gonner, Common Land and Enclosure, p. 156 n. [479] e.g. Nottingham Records, vol. iv. pp. 170–171, Nov. 4, 1577: “Any burgess that hath not been prentice to pay £10 and no pardon. Records of Leicester, vol. iii. p. 351, Oct. 17, 1598: “He is inhibited from dwelling in your corporation unless he finds bonds for £200 that neither his wife nor children shall be burdensome to the town." Southampton Court Leet Records, vol. i., Part I.: “One William Dye, undertenant to John Netley, dothe lyve idelly and hathe no trade.... He hathe 4 or 5 children in places from whence
- 80. he came whom he will bring shortly hither, yf he may be suffered here to remayne, whom we desyer may be examined and removed from hence according to the Statute.” [480] Some instances are given by Leonard, Early History of English Poor Relief, pp. 107–109. [481] Nottingham Records, vol. iv. pp. 304–307. [482] 31 Eliz. c. 7. [483] 35 Eliz. c. 6. [484] 39 Eliz. c. 3. [485] For petitions on this subject see Hist. MSS. Com., Cd. 784, pp. 81–82 (Wiltshire). The Warwickshire Quarter Sessions were much occupied with this, e.g. the following: “Trinity Sessions 1625. Fforasmuch as this Court was this present day informed ... by Sir Edward Marrowe, kt., and Thomas Ashley as the lords of the manor of Woolvey in this county ... that the said lords are content that William Wilcox of Woolvey in this countie shall build and erect a cottage for hys habitation hys wyfe and his small children uppon the waste within the said lordshippe, it is therefore ordered that the same being with consent of the lord as aforesaid that the same cottage shall be and continue,” and later “which cottage the Court doth licence" (Warwick Quarter Sessions MSS. Records). [486] “Considerations Concerning Common Fields and Enclosures,” Pseudonismus, 1654. [487] Moore, The Crying Sin of England in not Caring for the Poor: “And now alas, saith the poor cottier, there is no work for me, I must go where I may get my living. And hence it comes to pass that the open fielden towns have above double the number of cottiers they had wont to have, so that they cannot live one by another, and so put the fielden towns to vast expense, in caring for these poor that these enclosures have made.” [488] e.g. Hist. MSS. Com., Cd. 784, p. 95 (Wiltshire), pp. 292 and 298 (Worcester). [489] See Appendix I., No. VI. Miss Leonard (Trans. Royal Hist. Soc., vol. xix.) prints this document as referring to Norfolk, which appears to be an error.
- 81. [490] D'Ewes' Journal. Speech of Cecil, 1597.
- 82. CHAPTER IIITHE QUESTION OF TENANT RIGHT (a) The Tenants at Will and the LeaseholdersToC We have said above that we cannot measure the extent of the depopulation caused by enclosure, even for those years with regard to which figures are supplied us by Royal Commissions. But, after all, it is happily less important to arrive at an exact statistical estimate of the acres enclosed and of the number of tenants displaced, than it is to get a general view of the economic forces at work and of the structure of legal relationships upon which they operated. Given the economic reasons for the consolidation of holdings which were dominant in the sixteenth century, they could hardly have failed to result in evictions on a considerable scale, unless the tenants themselves had sufficient legal security to hold their own. If they had such security, the statistical analysis of displacements given above will fall into line with the general situation and be a valuable comment upon it. If they had not, then the figures, while a useful guide to the imagination, may stand when they confirm, but hardly when they contradict, the picture given by contemporaries. The accounts of the latter, though still not freed from the charge of exaggeration, will be supported by what we know of the general disposition of economic and legal forces. They probably heighten the colour and sharpen the outlines, but their indication of tendencies will be correct. In discussing the position of the small cultivator in the sixteenth century it was pointed out above that similarity of legal status was compatible with the greatest economic variety, and in considering their ability to resist attempted eviction it is essential to remember the converse truth, that tenants who were economically in a similar
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