168 BELL SYSTEM TECHNICAL JOURNAL 



Variations. Thus we get the so-called principle of least action, which can be 

 stated as follows: 



The particle moves, in a static field of force of the type (4), and with the 

 prescribed total energy E, in such a curve that the value of the integral 



r^^^' ([1 4- x[' + oc':r-[{E - Vfc-' - mlc'\"' + A, + A,x, + ^3:^:3) dx,, 



with the limits of integration held fixed, is stationary with respect to infinitesimal 

 variations of the trajectory which leave the end points unaltered. 



We have a precisely similar principle in Newtonian dynamics, but here 

 the integral in question is 



f " ' ([1 + X2 + x7\"\2m,{E - V)r- + Ax + A^x'^ + ^3^3) dx,. 



The last two integrals can be written more symmetrically, but not quite 

 so explicitly, as follows: 



r^ (\fi? T/\2 -2 2 2ii/2 I . dxi A dx2 . . dx^ , 



j^^[[iE-V)c -m^c] +M-^ + A,- + A,-ys, 



/:•(' 



,1/2 , ^ dxi , ^ dx2 , ^ dx?, 



[2m,{E - V)X" + ^^ ^ + ^2 ^' + A^ ^1 ds, 

 where Pi and P2 denote the end points of the trajectory, and ds' = dxi + 



dX2 + ^-^"3- 



VIII. The Hamilton-Jacobi Theory 

 Let us write 



W= L[xi(t),X2{t),x^{t),x[it),X2it),xUt),t]dL (23) 



hi 



We have already studied the variation of IF when /i and to are held fixed, 

 and the functions ;v„(/) are varied in such a way that the terminal values are 

 unaltered; and we have shown that under these circumstances the variation 

 of W vanishes, to the first order of small quantities, in the natural motion. ^- 

 In the following we shall study the variation of II' under some other 

 conditions. 



Specifically, we shall study the quantity All' defined by equation (23) 

 and the equation 



LMt) + Ut), ■■• , xs{t) + ^3(0, ^1 dt, 



• l + Ati 



12 I.e. a motion satisfying equations (1). 



