326 GENERALIZED COORDINATES 



This equation is true at every instant of the motion. Let us 

 integrate it between any two instants of the motion, say from 

 t = t : to t = 2 . We obtain 



(ufa + vjft + wfa)~ '* = f *S (T - W) dt. (155) 



The displaced motion has so far been subject to no restrictions 

 except that the difference between it and the actual motion must 

 always remain small. Let us now introduce the further restriction 

 that at times ^ and t z the configurations in the displaced motion 

 are to be identical with those in the actual motion. The displaced 

 motion is now one in which the imaginary system starts in the 

 same configuration as the actual system at time t t l} swerves 

 from the course of the actual system from time ^ to time t 2 

 (because the actual system obeys Newton's laws, while the imagi- 

 nary system does not), and ultimately ends in the same position 

 as the actual system at time t 2 . 



In consequence of this restriction on the motion of the imagi- 

 nary system, we have at times t t and t 2 , 



and similar relations for the other particles. Thus 



and equation (155) reduces to 



C*S(T-W)dt = Q. (156) 



tA 



Here we have an equation which depends only on the amounts 

 of the kinetic and potential energies of the system, and not on the 

 mechanism of the system. We shall find that from this single equa- 

 tion we can determine the motion of all the known parts of the 

 system as soon as T and W are known, without any knowledge of 

 the mechanism of the unknown parts. 



