176 BELL SYSTEM TECHNICAL JOURNAL 



From this it follows that we have the Hamiltonian canonical equations 

 .n _ dH . _ _dH 



dlTn OX'^ 



in any coordinate system. 



We have already Seen how to state Hamilton's principle in terms of general 

 coordinates. 



In the relativistic case the principle of least action takes the form: 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 



(■t')2 /r J.."* j„"~li/2 j^" 



J(xl)i 



dx dx 

 dx^ dx^ 



[(E — Vfc ^ — mlc'f" + A,n -r:^jdx\ 



dx' 



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

 variations of the trajectory which leave the end points unaltered. The corre- 

 sponding form of the principle for the Newtonian case is obvious. 



We are now in a position to dispose very quickly of the problem of formu- 

 lating the Hamilton- Jacobi theory in terms of general curvilinear 

 coordinates. 



The general form of the Hamiltonian function being given by (41) (for 

 the relativistic case) or (41') (for the Newtonian case), we can at once write 

 down the partial differential equation 



^ + H(x\ x\ x\ dW/dx\ dW/dx-, dW/dx\ t) = 0. (45) 



dt 



Let 



W = W{x\ x\ x\ t, a\ a\ a^) 



represent the complete solution of (45), without the irrelevant additive 

 constant of integration. 



Our chief problem is that of proving that the functions .v"(/, a^, a.-, a^, 

 /3i , /?2 , 183), TTnit, q:\ a'^, a^, ft , ft , 1S3) determined by the equations 



a^ ~ ^" ' ax" " """ ' 



where the /3's are further arbitrary- constants, satisfy the canonical equations 

 .„ _dll . _ _^^ 



dlTn a.V-" 



Now, referring to the proof given in Section VHI for the special case of 

 rectangular coordinates, we see at once that nothing in the proof depends 



