120 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



that the generalized component of momentum with regard to the 

 angular coordinate cp is the moment of momentum of the particle 

 [cf. 8, 23), 24)]. Systems in which there are coordinates having 

 the property that their non-momental part of the kinetic reaction 

 vanishes have peculiar properties, and are treated in 48. 



If we perform the differentiation of p s by the time, differentiating 

 equation 53), and remembering that the $'s depend only on the #'s, 

 we find for the momental part of the effective force 



~ = 



67) 





of which the first line, which we will call FJ> 1 \ is a linear function 

 of the generalized accelerations $ '. Here again our generalized 

 coordinates differ from rectangular, in that there is a part of the 

 momental force which is independent of the accelerations #", but 

 which is a homogeneous quadratic function of the velocities, 



r=m t=m o 



58) 



Consequently if at any instant of the motion we can change the signs 

 of all the velocities, and at the same time of all the accelerations, 

 the accelerational part of the momental force F^ will change its 

 sign, while the non- accelerational part F,W will be unchanged. We 

 may thus experimentally discriminate between the two. 



Effecting the differentiation in the case of the non-momental 

 force, we find 



which is also a homogeneous quadratic function of the velocities, 

 and thus possesses similar properties to Ff?\ Thus it is difficult to 

 discriminate experimentally between these two, unless we have some 

 experimental means of recognizing when the momentum p s remains 

 constant. In the simple example which we have used above, since 



the non -accelerational part of the momental force belonging to r 

 disappears, while the centrifugal or non-momental does not, while for cp 



