PROBLEM 2. Determine the Poisson brackets formed from the components of M.
SOLUTION. A direct calculation from formula (42.5) gives [Mx, My] = -M2, [My, M]
= -Mx, [Mz, Mx] = -My.
Since the momenta and co-ordinates of different particles are mutually independent variables,
it is easy to see that the formulae derived in Problems 1 and 2 are valid also for the total
momentum and angular momentum of any system of particles.
PROBLEM 3. Show that [, M2] = 0, where is any function, spherically symmetrical
about the origin, of the co-ordinates and momentum of a particle.
SOLUTION. Such a function can depend on the components of the vectors r and p only
through the combinations r2, p2, r. p. Hence
and similarly for The required relation may be verified by direct calculation from
formula (42.5), using these formulae for the partial derivatives.
PROBLEM 4. Show that [f, M] = n xf, where f is a vector function of the co-ordinates
and momentum of a particle, and n is a unit vector parallel to the z-axis.
SOLUTION. An arbitrary vector f(r,p) may be written as f = where
01, O2, 03 are scalar functions. The required relation may be verified by direct calculation
from formulae (42.9), (42.11), (42.12) and the formula of Problem 3.
$43. The action as a function of the co-ordinates
In formulating the principle of least action, we have considered the integral
(43.1)
taken along a path between two given positions q(1) and q(2) which the system
occupies at given instants t1 and t2. In varying the action, we compared the
values of this integral for neighbouring paths with the same values of q(t1)
and q(t2). Only one of these paths corresponds to the actual motion, namely
the path for which the integral S has its minimum value.
Let us now consider another aspect of the concept of action, regarding S
as a quantity characterising the motion along the actual path, and compare
the values of S for paths having a common beginning at q(t1) = q(1), but
passing through different points at time t2. In other words, we consider the
action integral for the true path as a function of the co-ordinates at the upper
limit of integration.
The change in the action from one path to a neighbouring path is given
(if there is one degree of freedom) by the expression (2.5):
8S =
Since the paths of actual motion satisfy Lagrange's equations, the integral
in 8S is zero. In the first term we put Sq(t1) = 0, and denote the value of
§43
The action as a function of the co-ordinates
139
8q(t2) by 8q simply. Replacing 0L/dq by p, we have finally 8S = pdq or, in
the general case of any number of degrees of freedom,
ES==Pisqu-
(43.2)
From this relation it follows that the partial derivatives of the action with
respect to the co-ordinates are equal to the corresponding momenta:
=
(43.3)
The action may similarly be regarded as an explicit function of time, by
considering paths starting at a given instant t1 and at a given point q(1), and
ending at a given point q(2) at various times t2 = t. The partial derivative
asiat thus obtained may be found by an appropriate variation of the integral.
It is simpler, however, to use formula (43.3), proceeding as follows.
From the definition of the action, its total time derivative along the path is
dS/dt = L.
(43.4)
Next, regarding S as a function of co-ordinates and time, in the sense des-
cribed above, and using formula (43.3), we have
dS
A comparison gives asid = L- or
(43.5)
Formulae (43.3) and (43.5) may be represented by the expression
(43.6)
for the total differential of the action as a function of co-ordinates and time
at the upper limit of integration in (43.1). Let us now suppose that the co-
ordinates (and time) at the beginning of the motion, as well as at the end,
are variable. It is evident that the corresponding change in S will be given
by the difference of the expressions (43.6) for the beginning and end of the
path, i.e.
dsp
(43.7)
This relation shows that, whatever the external forces on the system during
its motion, its final state cannot be an arbitrary function of its initial state;
only those motions are possible for which the expression on the right-hand
side of equation (43.7) is a perfect differential. Thus the existence of the
principle of least action, quite apart from any particular form of the Lagran-
gian, imposes certain restrictions on the range of possible motions. In parti-
cular, it is possible to derive a number of general properties, independent
of the external fields, for beams of particles diverging from given points in
140
The Canonical Equations
