§44
Maupertuis' principle
141
Writing the action in the form (43.8) and again replacing H by E, we have
(44.3)
The first term in this expression,
(44.4)
is sometimes called the abbreviated action.
Substituting (44.3) in (44.2), we find that
8S0=0.
(44.5)
Thus the abbreviated action has a minimum with respect to all paths which
satisfy the law of conservation of energy and pass through the final point
at any instant. In order to use such a variational principle, the momenta
(and so the whole integrand in (44.4)) must be expressed in terms of the
co-ordinates q and their differentials dq. To do this, we use the definition of
momentum:
(44.6)
and the law of conservation of energy:
E(g)
(44.7)
Expressing the differential dt in terms of the co-ordinates q and their differen-
tials dq by means of (44.7) and substituting in (44.6), we have the momenta
in terms of q and dq, with the energy E as a parameter. The variational prin-
ciple so obtained determines the path of the system, and is usually called
Maupertuis' principle, although its precise formulation is due to EULER and
LAGRANGE.
The above calculations may be carried out explicitly when the Lagrangian
takes its usual form (5.5) as the difference of the kinetic and potential energies:
The momenta are
and the energy is
The last equation gives
dt
(44.8)
142
The Canonical Equations
§44
substituting this in
Epides
we find the abbreviated action:
(44.9)
In particular, for a single particle the kinetic energy is T = 1/2 m(dl/dt)2,
where m is the mass of the particle and dl an element of its path; the variational
principle which determines the path is
${/[2m(B-U)]dl=0
(44.10)
where the integral is taken between two given points in space. This form is
due to JACOBI.
In free motion of the particle, U = 0, and (44.10) gives the trivial result
8 I dl = 0, i.e. the particle moves along the shortest path between the two
given points, i.e. in a straight line.
Let us return now to the expression (44.3) for the action and vary it with
respect to the parameter E. We have
substituting in (44.2), we obtain
(44.11)
When the abbreviated action has the form (44.9), this gives
=
(44.12)
which is just the integral of equation (44.8). Together with the equation of
the path, it entirely determines the motion.
PROBLEM
Derive the differential equation of the path from the variational principle (44.10).
SOLUTION. Effecting the variation, we have
f
In the second term we have used the fact that dl2 = dr2 and therefore dl d8l = dr. d&r.
Integrating this term by parts and then equating to zero the coefficient of Sr in the integrand,
we obtain the differential equation of the path:
§45
Canonical transformations
143
Expanding the derivative on the left-hand side and putting the force F = - auld gives
d2r/dl2=[F-(F.t)t]/2(E-U),
where t = dr/dl is a unit vector tangential to the path. The difference F-(F. t)t is the com-
ponent Fn of the force normal to the path. The derivative d2r/dl2 = dt/dl is known from
differential geometry to be n/R, where R is the radius of curvature of the path and n the unit
vector along the principal normal. Replacing E-U by 1mv2, we have (mv2/R)n = Fn, in
agreement with the familar expression for the normal acceleration in motion in a curved
path.
$45. Canonical transformations
The choice of the generalised co-ordinates q is subject to no restriction;
they may be any S quantities which uniquely define the position of the system
in space. The formal appearance of Lagrange's equations (2.6) does not
depend on this choice, and in that sense the equations may be said to be
invariant with respect to a transformation from the co-ordinates q1, q2,
to any other independent quantities Q1, Q2, The new co-ordinates Q are
functions of q, and we shall assume that they may explicitly depend on the
time, i.e. that the transformation is of the form
Qi=Qi(q,t)
(45.1)
(sometimes called a point transformation).
Since Lagrange's equations are unchanged by the transformation (45.1),
Hamilton's equations (40.4) are also unchanged. The latter equations, how-
ever, in fact allow a much wider range of transformations. This is, of course,
because in the Hamiltonian treatment the momenta P are variables inde-
pendent of and on an equal footing with the co-ordinates q. Hence the trans-
formation may be extended to include all the 2s independent variables P
and q:
Qt=Qi(p,q,t),
Pi = Pi(p, q,t).
(45.2)
This enlargement of the class of possible transformations is one of the im-
portant advantages of the Hamiltonian treatment.
The equations of motion do not, however, retain their canonical form
under all transformations of the form (45.2). Let us derive the conditions
which must be satisfied if the equations of motion in the new variables P, Q
are to be of the form
(45.3)
with some Hamiltonian H'(P,Q). When this happens the transformation is
said to be canonical.
The formulae for canonical transformations can be obtained as follows. It
has been shown at the end of §43 that Hamilton's equations can be derived
from the principle of least action in the form
(45.4)
144
The Canonical Equations
§45
in which the variation is applied to all the co-ordinates and momenta inde-
pendently. If the new variables P and Q also satisfy Hamilton's equations,
the principle of least action
0
(45.5)
must hold. The two forms (45.4) and (45.5) are equivalent only if their inte-
grands are the same apart from the total differential of some function F of
co-ordinates, momenta and time.t The difference between the two integrals
is then a constant, namely the difference of the values of F at the limits of
integration, which does not affect the variation. Thus we must have
=
Each canonical transformation is characterised by a particular function F,
called the generating function of the transformation.
Writing this relation as
(45.6)
we see that
Pi = 0F/dqi, =-0F/JQi H' = H+0F/dt;
(45.7)
here it is assumed that the generating function is given as a function of the
old and new co-ordinates and the time: F = F(q, Q, t). When F is known,
formulae (45.7) give the relation between p, q and P, Q as well as the new
Hamiltonian.
It may be convenient to express the generating function not in terms of the
variables q and Q but in terms of the old co-ordinates q and the new momenta
P. To derive the formulae for canonical transformations in this case, we must
effect the appropriate Legendre's transformation in (45.6), rewriting it as
=
The argument of the differential on the left-hand side, expressed in terms of
the variables q and P, is a new generating function (q, P, t), say. Thent
= Qi = ID/OPi, H' = H+d
(45.8)
We can similarly obtain the formulae for canonical transformations in-
volving generating functions which depend on the variables P and Q, or
p and P.
t We do not consider such trivial transformations as Pi = api, Qi = qt,H' = aH, with a an
arbitrary constant, whereby the integrands in (45.4) and (45.5) differ only by a constant
factor.
+ If the generating function is = fi(q, t)Pi, where the ft are arbitrary functions, we
obtain a transformation in which the new co-ordinates are Q = fi(q, t), i.e. are expressed
in terms of the old co-ordinates only (and not the momenta). This is a point transformation,
and is of course a particular canonical transformation.
§45
Canonical transformations
145
The relation between the two Hamiltonians is always of the same form:
the difference H' - H is the partial derivative of the generating function with
respect to time. In particular, if the generating function is independent of
time, then H' = H, i.e. the new Hamiltonian is obtained by simply substitut-
ing for P, q in H their values in terms of the new variables P, Q.
The wide range of the canonical transformations in the Hamiltonian treat-
ment deprives the generalised co-ordinates and momenta of a considerable
part of their original meaning. Since the transformations (45.2) relate each
of the quantities P, Q to both the co-ordinates q and the momenta P, the
variables Q are no longer purely spatial co-ordinates, and the distinction
between Q and P becomes essentially one of nomenclature. This is very
clearly seen, for example, from the transformation Q = Pi, Pi = -qi,
which obviously does not affect the canonical form of the equations and
amounts simply to calling the co-ordinates momenta and vice versa.
On account of this arbitrariness of nomenclature, the variables P and q in
the Hamiltonian treatment are often called simply canonically conjugate
quantities. The conditions relating such quantities can be expressed in terms
of Poisson brackets. To do this, we shall first prove a general theorem on the
invariance of Poisson brackets with respect to canonical transformations.
Let [f,g]p,a be the Poisson bracket, for two quantities f and g, in which
the differentiation is with respect to the variables P and q, and [f,g]p,Q that
in which the differentiation is with respect to P and Q. Then
(45.9)
The truth of this statement can be seen by direct calculation, using the for-
mulae of the canonical transformation. It can also be demonstrated by the
following argument.
First of all, it may be noticed that the time appears as a parameter in the
canonical transformations (45.7) and (45.8). It is therefore sufficient to prove
(45.9) for quantities which do not depend explicitly on time. Let us now
formally regard g as the Hamiltonian of some fictitious system. Then, by
formula (42.1), [f,g]p,a = df/dt. The derivative df/dt can depend only on
the properties of the motion of the fictitious system, and not on the particular
choice of variables. Hence the Poisson bracket [f,g] is unaltered by the
passage from one set of canonical variables to another.
Formulae (42.13) and (45.9) give
[Qi, Qk]p,a = 0, [Pi,Pk]p,a = 0,
(45.10)
These are the conditions, written in terms of Poisson brackets, which must
be satisfied by the new variables if the transformation P, q P, Q is canonical.
It is of interest to observe that the change in the quantities P, q during the
motion may itself be regarded as a series of canonical transformations. The
meaning of this statement is as follows. Let qt, Pt be the values of the canonical
t Whose generating function is
6*
146
The Canonical Equations
§46
variables at time t, and qt+r, Pt+r their values at another time t +T. The latter
are some functions of the former (and involve T as a parameter):
If these formulae are regarded as a transformation from the variables Qt, Pt
to qt+r, Pttr, then this transformation is canonical. This is evident from the
expression ds = for the differential of the action S(qt++,
qt) taken along the true path, passing through the points qt and qt++ at given
times t and t + T (cf. (43.7)). A comparison of this formula with (45.6) shows
that - S is the generating function of the transformation.
46. Liouville's theorem
For the geometrical interpretation of mechanical phenomena, use is often
made of phase space. This is a space of 2s dimensions, whose co-ordinate axes
correspond to the S generalised co-ordinates and S momenta of the system
concerned. Each point in phase space corresponds to a definite state of the
system. When the system moves, the point representing it describes a curve
called the phase path.
The product of differentials dT = dq1 ... dqsdp1 dps may be regarded
as an element of volume in phase space. Let us now consider the integral
I dT taken over some region of phase space, and representing the volume of
that region. We shall show that this integral is invariant with respect to
canonical transformations; that is, if the variables P, q are replaced by
P, Q by a canonical transformation, then the volumes of the corresponding
regions of the spaces of P, and P, Q are equal:
...dqsdp1...dps =
(46.1)
The transformation of variables in a multiple integral is effected by the
formula I .jdQ1...dQsdP1...dPz = S... I Ddq1 dp1...dps,
where
(46.2)
is the Jacobian of the transformation. The proof of (46.1) therefore amounts
to proving that the Jacobian of every canonical transformation is unity:
D=1.
(46.3)
We shall use a well-known property of Jacobians whereby they can be
treated somewhat like fractions. "Dividing numerator and denominator" by
0(91, ..., qs, P1, Ps), we obtain
Another property of Jacobians is that, when the same quantities appear in
both the partial differentials, the Jacobian reduces to one in fewer variables,
§47
The Hamilton-Jacobi equation
147
in which these repeated quantities are regarded as constant in carrying out
the differentiations. Hence
(46.4)
P=constant
q=constant
The Jacobian in the numerator is, by definition, a determinant of order s
whose element in the ith row and kth column is Representing the
canonical transformation in terms of the generating function (q, P) as in
(45.8), we have = In the same way we find that the
ik-element of the determinant in the denominator of (46.4) is
This means that the two determinants differ only by the interchange of rows
and columns; they are therefore equal, so that the ratio (46.4) is equal to
unity. This completes the proof.
Let us now suppose that each point in the region of phase space considered
moves in the course of time in accordance with the equations of motion of the
mechanical system. The region as a whole therefore moves also, but its volume
remains unchanged:
f dr = constant.
(46.5)
This result, known as Liouville's theorem, follows at once from the invariance
of the volume in phase space under canonical transformations and from the
fact that the change in p and q during the motion may, as we showed at the end
of §45, be regarded as a canonical transformation.
In an entirely similar manner the integrals
11 2 dae dph
,
in which the integration is over manifolds of two, four, etc. dimensions in
phase space, may be shown to be invariant.
47. The Hamilton-Jacobi equation
In §43 the action has been considered as a function of co-ordinates and
time, and it has been shown that the partial derivative with respect to time
of this function S(q, t) is related to the Hamiltonian by
and its partial derivatives with respect to the co-ordinates are the momenta.
Accordingly replacing the momenta P in the Hamiltonian by the derivatives
as/aq, we have the equation
(47.1)
which must be satisfied by the function S(q, t). This first-order partial
differential equation is called the Hamilton-Jacobi equation.
148
The Canonical Equations
§47
Like Lagrange's equations and the canonical equations, the Hamilton-
Jacobi equation is the basis of a general method of integrating the equations
of motion.
Before describing this method, we should recall the fact that every first-
order partial differential equation has a solution depending on an arbitrary
function; such a solution is called the general integral of the equation. In
mechanical applications, the general integral of the Hamilton-Jacobi equation
is less important than a complete integral, which contains as many independent
arbitrary constants as there are independent variables.
The independent variables in the Hamilton-Jacobi equation are the time
and the co-ordinates. For a system with s degrees of freedom, therefore, a
complete integral of this equation must contain s+1 arbitrary constants.
Since the function S enters the equation only through its derivatives, one
of these constants is additive, so that a complete integral of the Hamilton-
Jacobi equation is
Sft,q,saas)+
(47.2)
where X1, ..., as and A are arbitrary constants.
Let us now ascertain the relation between a complete integral of the
Hamilton-Jacobi equation and the solution of the equations of motion which
is of interest. To do this, we effect a canonical transformation from the
variables q, P to new variables, taking the function f (t, q; a) as the
generating function, and the quantities a1, A2, ..., as as the new momenta.
Let the new co-ordinates be B1, B2, ..., Bs. Since the generating function
depends on the old co-ordinates and the new momenta, we use formulae
(45.8): Pi = af/dqi, Bi = af/dar, H' = H+dfdd. But since the function f
satisfies the Hamilton-Jacobi equation, we see that the new Hamiltonian is
zero: H' = H+af/dt = H+as/t = 0. Hence the canonical equations in
the new variables are di = 0, Bi = 0, whence
ay=constant,
Bi = constant.
(47.3)
By means of the S equations af/dai = Bi, the S co-ordinates q can be expressed
in terms of the time and the 2s constants a and B. This gives the general
integral of the equations of motion.
t Although the general integral of the Hamilton-Jacobi equation is not needed here, we
may show how it can be found from a complete integral. To do this, we regard A as an arbi-
trary function of the remaining constants: S = f(t, q1, ..., q8; a1, as) +A(a1, as). Re-
placing the Ai by functions of co-ordinates and time given by the S conditions asidar = 0,
we obtain the general integral in terms of the arbitrary function A(a1,..., as). For, when the
function S is obtained in this manner, we have
as
The quantities (as/dqs)a satisfy the Hamilton-Jacobi equation, since the function S(t, q; a)
is assumed to be a complete integral of that equation. The quantities asida therefore satisfy
the same equation.
§48
Separation of the variables
149
Thus the solution of the problem of the motion of a mechanical system by
the Hamilton-Jacobi method proceeds as follows. From the Hamiltonian,
we form the Hamilton-Jacobi equation, and find its complete integral (47.2).
Differentiating this with respect to the arbitrary constants a and equating
the derivatives to new constants B, we obtain S algebraic equations
asidar=Bt,
(47.4)
whose solution gives the co-ordinates q as functions of time and of the 2s
arbitrary constants. The momenta as functions of time may then be found
from the equations Pi = aslaqi.
If we have an incomplete integral of the Hamilton-Jacobi equation, depend-
ing on fewer than S arbitrary constants, it cannot give the general integral
of the equations of motion, but it can be used to simplify the finding of the
general integral. For example, if a function S involving one arbitrary con-
stant a is known, the relation asida = constant gives one equation between
q1, ..., qs and t.
The Hamilton-Jacobi equation takes a somewhat simpler form if the func-
tion H does not involve the time explicitly, i.e. if the system is conservative.
The time-dependence of the action is given by a term -Et:
S = So(g)-Et
(47.5)
(see 44), and substitution in (47.1) gives for the abbreviated action So(q)
the Hamilton-Jacobi equation in the form
(47.6)
$48. Separation of the variables
In a number of important cases, a complete integral of the Hamilton-
Jacobi equation can be found by "separating the variables", a name given to
the following method.
Let us assume that some co-ordinate, q1 say, and the corresponding
derivative asia appear in the Hamilton-Jacobi equation only in some
combination (q1, which does not involve the other co-ordinates, time,
or derivatives, i.e. the equation is of the form
(48.1)
where qi denotes all the co-ordinates except q1.
We seek a solution in the form of a sum:
(48.2)
150
The Canonical Equations
§48
substituting this in equation (48.1), we obtain
(48.3)
Let us suppose that the solution (48.2) has been found. Then, when it is
substituted in equation (48.3), the latter must become an identity, valid (in
particular) for any value of the co-ordinate q1. When q1 changes, only the
function is affected, and so, if equation (48.3) is an identity, must be a
constant. Thus equation (48.3) gives the two equations
(48.4)
= 0,
(48.5)
where a1 is an arbitrary constant. The first of these is an ordinary differential
equation, and the function S1(q1) is obtained from it by simple integration.
The remaining partial differential equation (48.5) involves fewer independent
variables.
If we can successively separate in this way all the S co-ordinates and the
time, the finding of a complete integral of the Hamilton-Jacobi equation is
reduced to quadratures. For a conservative system we have in practice to
separate only S variables (the co-ordinates) in equation (47.6), and when this
separation is complete the required integral is
(48.6)
where each of the functions Sk depends on only one co-ordinate; the energy
E, as a function of the arbitrary constants A1, As, is obtained by substituting
So = in equation (47.6).
A particular case is the separation of a cyclic variable. A cyclic co-ordinate
q1 does not appear explicitly in the Hamiltonian, nor therefore in the Hamilton-
Jacobi equation. The function (91, reduces to as/da simply, and
from equation (48.4) we have simply S1 = x1q1, so that
(48.7)
The constant a1 is just the constant value of the momentum P1 = asida
corresponding to the cyclic co-ordinate.
The appearance of the time in the term - Et for a conservative system
corresponds to the separation of the "cyclic variable" t.
Thus all the cases previously considered of the simplification of the integra-
tion of the equations of motion by the use of cyclic variables are embraced
by the method of separating the variables in the Hamilton-Jacobi equation.
To those cases are added others in which the variables can be separated even
though they are not cyclic. The Hamilton-Jacobi treatment is consequently
the most powerful method of finding the general integral of the equations of
motion.