41] METHOD OF HAMILTON -JACOB! 137 



The arbitrary constant of energy, li, takes the place of one of the 

 constants c. 



The variation of the action on changing the terminal configura- 

 tions will cause a change in the energy necessary, h, if the time of 

 transit, t t Q , is unchanged. Accordingly 



r==m r = m 1 



106) dA = dS + (t-tJ dh =p r dq r - C r dc r + (t- g dh, 



by equation 100). We have therefore 



107) p, - f|, 



108) -fc-V 



109) t - t. = |. 



As examples of the use of the method of Hamilton we will 

 now solve a few problems that have been already treated. 



First, let us take the case of any number of free particles. 

 We have 



Putting 



_dT_ 



Pxr fig,! m r x rj 



dT 

 110) Ptt jVk&t 



- - ' 



Pzr Of Wl r 8 



this becomes 



By 107) 



< ^ x dA dA 3 A 



P*r = ^> P,r = 3j- f > P>r= Wr 



and equation 105) then is 



In the case of a single particle comparing equations 110) and 



111) we have 



dA f dA 



w me ~w 



In other words if the action A is expressed in terms of the co- 

 ordinates x, y, s, the momentum of a particle describing any path 



