space. The study of these properties forms a part of the subject of geometrical
optics.+
It is of interest to note that Hamilton's equations can be formally derived
from the condition of minimum action in the form
(43.8)
which follows from (43.6), if the co-ordinates and momenta are varied inde-
pendently. Again assuming for simplicity that there is only one co-ordinate
and momentum, we write the variation of the action as
= dt - (OH/dp)8p dt].
An integration by parts in the second term gives
At the limits of integration we must put 8q = 0, so that the integrated term
is zero. The remaining expression can be zero only if the two integrands
vanish separately, since the variations Sp and 8q are independent and arbitrary
dq = (OH/OP) dt, dp = - (dH/dq) dt, which, after division by dt, are
Hamilton's equations.
$44. Maupertuis' principle
The motion of a mechanical system is entirely determined by the principle
of least action: by solving the equations of motion which follow from that
principle, we can find both the form of the path and the position on the path
as a function of time.
If the problem is the more restricted one of determining only the path,
without reference to time, a simplified form of the principle of least action
may be used. We assume that the Lagrangian, and therefore the Hamilton-
ian, do not involve the time explicitly, SO that the energy of the system is
conserved: H(p, q) = E = constant. According to the principle of least action,
the variation of the action, for given initial and final co-ordinates and times
(to and t, say), is zero. If, however, we allow a variation of the final time t,
the initial and final co-ordinates remaining fixed, we have (cf.(43.7))
8S = -Hot.
(44.1)
We now compare, not all virtual motions of the system, but only those
which satisfy the law of conservation of energy. For such paths we can
replace H in (44.1) by a constant E, which gives
SS+Est=0.
(44.2)
t See The Classical Theory of Fields, Chapter 7, Pergamon Press, Oxford 1962.
§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:
