Stationary Action.
The preceding theories give us statements applicable to the system at
any one instant of its motion. We now come to a series of theorems
relating to the whole motion of the system between any two
configurations through which it passes, viz. we consider the actual
motion and compare it with other imaginable motions, differing
infinitely little from it, between the same two configurations. We use
the symbol [delta] to denote the transition from the actual to any one
of the hypothetical motions.
The best-known theorem of this class is that of _Least Action_,
originated by P.L.M. de Maupertuis, but first put in a definite form
by Lagrange. The "action" of a single particle in passing from one
position to another is the space-integral of the momentum, or the
time-integral of the _vis viva_. The action of a dynamical system is
the sum of the actions of its constituent particles, and is
accordingly given by the formula
_ _ _
/ / /
A = [Sigma] | mvds = [Sigma] | mv squareddt = 2 | [Tau]dt. (1)
_/ _/ _/
The theorem referred to asserts that the free motion of a conservative
system between any two given configurations is characterized by the
property
[delta]A = 0, (2)
provided the total energy have the same constant value in the varied
motion as in the actual motion.
If t, t' be the times of passing through the initial and final
configurations respectively, we have
_
/ t'
[delta]A = [delta] | [Sigma]m(x` squared + y` squared + z` squared)dt
_/t
_
/ t'
= 2 | [delta][Tau]dt + 2[Tau]'[delta]t' + 2[Tau][delta]t, (3)
_/t
since the upper and lower limits of the integral must both be regarded
as variable. This may be written
_ _
/ t' / t'
[delta]A = | [delta][Tau]dt + | [Sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`)dt + 2[Tau]'[delta]t' - 2[Tau][delta]t
_/t _/t
_ _ _
/ t' | | t'
= | [delta][Tau]dt + | [Sigma]m (x`[delta]x + y`[delta]y + z`[delta]z |
_/t |_ _| t
_
/ t'
- | [Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z)dt + 2[Tau]'[delta]t' - 2[Tau][delta]t. (4)
_/t
Now, by d'Alembert's principle,
[Sigma]m( x"[delta]x + y"[delta]y + z"[delta]z ) = -[delta]V, (5)
and by hypothesis we have
[delta]([Tau] + V) = 0. (6)
The formula therefore reduces to
_ _
| |t'
[delta]A = | [Sigma]m (x`[delta]x + y`[delta]y + z`[delta]z) | + 2[Tau]'[delta]t' - 2[Tau][delta]t. (7)
|_ _|t
Since the terminal configurations are unaltered, we must have at the
lower limit
[delta]x + x`[delta]t = 0, [delta]y + y`[delta]t = 0, [delta]z + z`[delta]t = 0, (8)
with similar relations at the upper limit. These reduce (7) to the
form (2).
The equation (2), it is to be noticed, merely expresses that the
variation of A vanishes _to the first order_; the phrase _stationary
action_ has therefore been suggested as indicating more accurately
what has been proved. The action in the free path between two given
configurations is in fact not invariably a minimum, and even when a
minimum it need not be the _least possible_ subject to the given
conditions. Simple illustrations are furnished by the case of a single
particle. A particle moving on a smooth surface, and free from
extraneous force, will have its velocity constant; hence the theorem
in this case resolves itself into
_
/
[delta] | ds = 0, (9)
_/
i.e. the path must be a geodesic line. Now a geodesic is not
necessarily the _shortest_ path between two given points on it; for
example, on the sphere a great-circle arc ceases to be the shortest
path between its extremities when it exceeds 180 deg.. More generally,
taking any surface, let a point P, starting from O, move along a
geodesic; this geodesic will be a minimum path from O to P until P
passes through a point O' (if such exist), which is the intersection
with a consecutive geodesic through O. After this point the minimum
property ceases. On an anticlastic surface two geodesics cannot
intersect more than once, and each geodesic is therefore a minimum
path between any two of its points. These illustrations are due to
K.G.J. Jacobi, who has also formulated the general criterion,
applicable to all dynamical systems, as follows:--Let O and P denote
any two configurations on a natural path of the system. If this be the
sole free path from O to P with the prescribed amount of energy, the
action from O to P is a minimum. But if there be several distinct
paths, let P vary from coincidence with O along the first-named path;
the action will then cease to be a minimum when a configuration O' is
reached such that two of the possible paths from O to O' coincide. For
instance, if O and P be positions on the parabolic path of a
projectile under gravity, there will be a second path (with the same
energy and therefore the same velocity of projection from O), these
two paths coinciding when P is at the other extremity (O', say) of the
focal chord through O. The action from O to P will therefore be a
minimum for all positions of P short of O'. Two configurations such as
O and O' in the general statement are called conjugate _kinetic foci_.
Cf. VARIATIONS, CALCULUS OF.
Before leaving this topic the connexion of the principle of stationary
action with a well-known theorem of optics may be noticed. For the
motion of a particle in a conservative field of force the principle
takes the form
_
/
[delta] | vds = 0. (10)
_/
On the corpuscular theory of light v is proportional to the refractive
index mu of the medium, whence
_
/
[delta] | muds = 0. (11)
_/
Hamiltonian principle.
In the formula (2) the energy in the hypothetical motion is
prescribed, whilst the time of transit from the initial to the final
configuration is variable. In another and generally more convenient
theorem, due to Hamilton, the time of transit is prescribed to be the
same as in the actual motion, whilst the energy may be different and
need not (indeed) be constant. Under these conditions we have
_
/t'
[delta] | ([Tau] - V)dt = 0, (12)
_/t
where t, t' are the prescribed times of passing through the given
initial and final configurations. The proof of (12) is simple; we have
_ _ _
/t' /t' /t'
[delta] | ([Tau] - V)dt = | ([delta][Tau] - [delta]V)dt = | {[Sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`) - [delta]V}dt
_/t _/t _/t
_ _
| |t'
= | [Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) |
|_ _|t
_
/t'
- | {[Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) + [delta]V}dt (13)
_/t
The integrated terms vanish at both limits, since by hypothesis the
configurations at these instants are fixed; and the terms under the
integral sign vanish by d'Alembert's principle.
The fact that in (12) the variation does not affect the time of
transit renders the formula easy of application in any system of
co-ordinates. Thus, to deduce Lagrange's equations, we have
_ _
/t' /t'/dP[Tau] dP[Tau] dPV \
| ([delta][Tau]-[delta]V)dt = | ( -------[delta]q`1 + -------[delta]q1 + ... - ----[delta]q1 - ...)dt
_/t _/t \ dPq`1 dPq1 dPq1 /
_ _
| |t'
= | p1[delta]q1 + p2[delta]q2 + ... |
|_ _|t
_ _ _
/t'| / dP[Tau] dPV \ / dP[Tau] dPV\ |
- | | (p`1 - ------- + ---- )[delta]q1 + (p`2 - ------- + -----)[delta]q2 + ...|dt. (14)
_/t |_ \ dPq1 dPq1/ \ dPq2 dPq2/ _|
The integrated terms vanish at both limits; and in order that the
remainder of the right-hand member may vanish it is necessary that the
coefficients of [delta]q1, [delta]q2, ... under the integral sign
should vanish for all values of t, since the variations in question
are independent, and subject only to the condition of vanishing at the
limits of integration. We are thus led to Lagrange's equation of
motion for a conservative system. It appears that the formula (12) is
a convenient as well as a compact embodiment of the whole of ordinary
dynamics.
Extension to cyclic systems.
The modification of the Hamiltonian principle appropriate to the case
of cyclic systems has been given by J. Larmor. If we write, as in Sec. 1
(25),