﻿Partition of Energy and Newtonian Mechanics. 385 



of the system between the two instants is presumed to be 

 governed by the ordinary Newtonian system of mechanical 

 laws, so that a general set of coordinates sufficient for our 

 purposes would be provided in any set or generalized 

 Lagrangian coordinates of the system p l5 p 2 , • • • Vn-> anc ^ * ne 

 momenta corresponding to these, say q^ q 2i . . . g n . The 

 equations of motion can be taken in any of the usual general 

 forms. 



If we construct a 2rc-dimensional space a single point 

 in this space, namely the point whose coordinates are 

 Pi? p 2 , ... p n ; </ l5 q 2 , . . . q n , will represent the state of the 

 system at any instant, and the general equations of motion 

 are the equations to the paths or trajectories traced out in 

 this space by the representative points as they follow out 

 the different possible motions of the system. It is obvious 

 that through every point in the generalized space there is 

 one and only one trajectory, and that as a point moves along 

 a trajectory and so follows the motion of a system, its 

 velocity at any point depends only on the coordinates of the 

 point and not on the time. 



In the usual manner we can therefore imagine every 

 region of the generalized space which represents a physically 

 possible state of the system to be filled with so many repre- 

 sentative points, that the whole collection of points may be 

 regarded as forming a continuous fluid. The general equa- 

 tions of motion then assert that this fluid moves along fixed 

 stream-lines and that the velocity at any point is constant. 

 Moreover, we know from Liouville's theorem that if we 

 follow the motion of all the points from the inside of any 

 elementary parallelopiped 



d Pl ■ d l\ • • • dp nQ . dq h . dq 2Q .... dq nQ 



at time t , they will be found at time t in the corresponding 

 parallelopiped 



dpi . dp - 2 . . . dp n . dq x . dq 2 . . . dq n , 



and the volumes of these parallelopipeds are the same. The 

 same is also true of the projections of the points on the 

 elementary area (dp r dq t ) parallel to one of the coordinate 

 planes defined by a generalized geometrical coordinate and 

 its corresponding momentum, the area remaining constant. 



The density of the fluid, or the density of the aggregation 

 of the representative points thus remains constant through- 

 out all time, so also does its density parallel to any {p r q r ^ 

 plane. The initial distribution of the density is entirely at 



Phil Mag. S. 6. Vol. 29. No. 171. March 1915. 2 C ' 



