age of

energy doubled.

Time aver cording to the unit generally adopted, the action of a system which has not varied in its kinetic energy, is twice the amount of the energy multiplied by the time from the epoch. Or if the energy has been sometimes greater and sometimes less, the action at time t, is the double of what we may call the time-integral of the energy, that is to say, is denoted in the integral calculus by

2 Tdr

0

where T denotes the kinetic energy at any time, between the epoch and t.

Let m be the mass, and v the velocity at time 7, of any one of the material points of which the system is composed. We have

This may be put otherwise by taking ds to denote the space described by a particle in time dr, so that vdr=ds, and therefore

A=/Σmvds

(3),

Space average of momenta.

Least action.

or, if x, y, z be the rectangular co-ordinates of m at any time,

Hence we might, as many writers in fact have virtually done, define action thus:

The action of a system is equal to the sum of the average momenta for the spaces described by the particles from any era each multiplied by the length of its path.

319. The principle of Least Action is this:-Of all the different sets of paths along which a conservative system may be guided to move from one configuration to another, with the sum of its potential and kinetic energies equal to a given constant, that one for which the action is the least is such that the system will require only to be started with the proper velocities, to move along it unguided.

Let x, y, z be the co-ordinates of a particle, m, of the system, at time 7, and V the potential energy of the system in its particular configuration at this instant; and let it be required to find the way to pass from one given configuration to another with velocities at each instant satisfying the condition

Σ }m(x2+ÿ2+ż2)+V=E, a constant

(5).

so that A, or

2m(xdx+ydy+żdz)

may be the least possible.

By the method of variations we must have SA=0, where

SA=2m(xdsx+ÿdôy+żdôz+dxdx+8ÿdy+8żdz)

(6).

Taking in this dx=xdτ, dy=ýdτ, dz=żdт, and remarking that

we have

Σm(x8x+ÿồÿ+żdż)=8T

ƒ 2m(&xdx+ôýdy+ôżdz)= [* 8Tdr

Also, by integration by parts,

(7),

(8).

Least action.

ƒ£m(¿đôx+.....)= {Σm(x8x+.....)} −[Σm(x8x+.....)]—ƒΣm(x8x+.......)dτ,
where [...] and {...} denote the values of the quantities enclosed,
at the beginning and end of the motion considered, and where,
further, it must be remembered that dx=xdτ, etc. Hence,
from above,

81= 2m (các+gây +202)} [(20x+yy+8z)]

+ [' dr[8T— 2m(ï&x+ÿöy+żôz)]

(9).

This, it may be observed, is a perfectly general kinematical expres-
sion, unrestricted by any terminal or kinetic conditions. Now
in the present problem we suppose the initial and final positions
to be invariable. Hence the terminal variations, dx, etc., must
all vanish, and therefore the integrated expressions {...}, [...] dis-
appear. Also, in the present problem 8T=-8V, by the equation
of energy (5). Hence, to make 84-0, since the intermediate
variations, &x, etc., are quite arbitrary, subject only to the con-
ditions of the system, we must have

which [(4), § 293 above] is the general variational equation of
motion of a conservative system. This proves the proposition.
Hence also it follows that

"stationary

Hamilton.

320. In any unguided motion whatever, of a 'conservative Why called system, the Action from any one stated position to any other, action" by though not necessarily a minimum, fulfils the stationary condition, that is to say, the condition that the variation vanishes, which secures either a minimum or maximum, or maximumminimum.

This can scarcely be made intelligible without mathematical language. Let (X1, Y1, Z1), (X, Y, Z), etc., be the co-ordinates

Stationary action.

Varying action.

of particles, m1, ma, etc., composing the system; at any time of the actual motion. Let V be the potential energy of the system, in this configuration; and let E denote the given value of the sum of the potential and kinetic energies. The equation of energy is

2

2

2

{{m1(x2+ÿ12+z12)+m1(x12+ÿì2+ż,2)+etc.}+V=E (1). Choosing any part of the motion, for instance that from time 0 to time t, we have, for the action during it,

A= ['"(E—V)dr=Et— ["var

(11). Let now the system be guided to move in any other way possible for it, with any other velocities, from the same initial to the same final configuration as in the given motion, subject only to the condition, that the sum of the kinetic and potential energies shall still be E. Let (x, y, z), etc., be the co-ordinates, and V the corresponding potential energy; and let (x, y, z), etc., be the component velocities, at time in this arbitrary motion; equation (2) still holding, for the accented letters, with only E unchanged. For the action we shall have

where t' is the time occupied by this supposed motion. Let now Ø denote a small numerical quantity, and let έ1, 71, etc., be finite

vanishes

The "principle of stationary action" is, that

when is made infinitely small, for every possible deviation
(0, 0, etc.) from the natural way and velocities, subject only
to the equation of energy and to the condition of passing through
the stated initial and final configurations: and conversely, that if
I'V
yanishes with for every possible such deviation, from a
Ө
certain way and velocities, specified by (x1, 31, 2,), etc., as the
co-ordinates at t, this way and these velocities are such that the
system unguided will move accordingly if only started with proper
velocities from the initial configuration.

321. From this principle of stationary action, founded, as we have seen, on a comparison between a natural motion, and any other motion, arbitrarily guided and subject only to the

action.

law of energy, the initial and final configurations of the Varying system being the same in each case; Hamilton passes to the consideration of the variation of the action in a natural or unguided motion of the system produced by varying the initial and final configurations, and the sum of the potential and kinetic energies. The result is, that

322. The rate of decrease of the action per unit of increase of any one of the free (generalized) co-ordinates (§ 204) specifying the initial configuration, is equal to the corresponding (generalized) component momentum [§ 313, (c)] of the actual motion from that configuration: the rate of increase of the action per unit increase of any one of the free co-ordinates specifying the final configuration, is equal to the corresponding component momentum of the actual motion towards this second configuration: and the rate of increase of the action per unit increase of the constant sum of the potential and kinetic energies, is equal to the time occupied by the motion of which the action is reckoned.

To prove this we must, in our previous expression (9) for 84,
now suppose the terminal co-ordinates to vary; ST to become
SE-SV, in which SE is a constant during the motion; and each
set of paths and velocities to belong to an unguided motion of
the system, which requires (10) to hold. Hence

SA={Zm(x&x+ÿôy+ż82)} −[Σm(xồx+yồy+żdz)]+18E (13).
If, now, in the first place, we suppose the particles constituting
the system to be all free from constraint, and therefore (x, y, z)
for each to be three independent variables, and if, for distinct-
ness, we denote by (x, y, z) and (1, 1, 1) the co-ordinates
of m, in its initial and final positions, and by (xí, ÿí, ż1'), (X1, Ý1, Z1)
the components of the velocity it has at those points, we have,
from the preceding, according to the ordinary notation of partial
differential coefficients,

In these equations we must suppose A to be expressed as a func-
tion of the initial and final co-ordinates, in all six times as many

Varying action.

independent variables as there are of particles; and E, one more variable, the sum of the potential and kinetic energies.

If the system consists not of free particles, but of particles connected in any way forming either one rigid body or any number of rigid bodies connected with one another or not, we might, it is true, be contented to regard it still as a system of free particles, by taking into account, among the impressed forces, the forces necessary to compel the satisfaction of the conditions of connexion. But although this method of dealing with a system of connected particles is very simple, so far as the law of energy merely is concerned, Lagrange's methods, whether that of “equations of condition," or, what for our present purposes is much more convenient, his "generalized co-ordinates," relieve us from very troublesome interpretations when we have to consider the displacements of particles due to arbitrary variations in the configuration of a system.

Let us suppose then, for any particular configuration (x1, y1, Z1) (X, Y, Z)..., the expression

m1(x18x1+ý1dy1+ż,dz,)+etc., to become έ84+784+88¥+etc. (15),

when transformed into terms of 4, 4, 0..., generalized co-ordinates, as many in number as there are of degrees of freedom for the system to move [§ 313, (c)].

The same transformation applied to the kinetic energy of the system would obviously give

η,

(16).

and hence έ, 7, §, etc., are those linear functions of the generalized velocities which we have designated as "generalized components of momentum;" and which, when T, the kinetic energy, is expressed as a quadratic function of the velocities (of course with, in general, functions of the co-ordinates 4, 4, 0, etc., for the coefficients) are derivable from it thus:

Hence, taking as before non-accented letters for the second, and accented letters for the initial, configurations of the system re