114 Force- Function in the Theory of Ordinary Relativity. 

 for a particle under no forces. If forces are acting on it, put 



72 



p ^2 fe y-> z > u ) = ( f*> U /« Q? • • • ( 4 ) 



■where //, is an invariant corresponding to the mass. 

 The momentum is 



(Px, Vy, p n Pu) = p> ^ 0, y, *, u) . . . (5) 



If the canonical form of the equations is possible, we have 



7TT dx . , dy , fife , du 1 



dR = Tt dj ' I+ dr dp > + It*' + ch d * 



\cIt dr J dr dr 



= y *(pI+pI +pI+pI) - (M* +fy d y + f zdz +M U )- (6) 



The first term in this is d(^/mc 2 ) = 0. Hence the condition 

 that H may exist is that f^dx +f y dy +f z dz + f u du shall be a 

 complete differential. Hence (f x ,ft,,fz, fu) is the gradient of 

 a scalar, U, say ; and U must be independent of the momenta. 

 In other words, the canonical form of the equations of motion 

 implies that a force-function exists. 



„ _._ , dx , dy . dz , du 



3. Next, f*-r+Jyi~ A rt*-T ~~t u T 



} ' dr • dr dr dr 



-*,%. ■ ■ ■ m 



*» r-i-J {&)'+(&*<$}. . . ,8, 



This is the equation of conservation of energy, in three- 

 dimensional motion. If it be multiplied by any function of 

 7, say R(y), it shows that the rate of increase of \/juc 2 yRdy 

 is equal to the scalar product of ^>{fz,fy,fx) and dx, dy, dz. 

 Hence if we use the terms "kinetic energy" to denote 

 ^fic 2 yRdy, and "force" for ~R(f x >fy,fz) we shall have the 

 relation 



Increase of kinetic energy = work done . . (9) 



whatever be the form of R. There is no way of deciding 

 between these various definitions of force and energy, the form 



