172 BELL SYSTEM TECHNICAL JOURNAL 



On the other hand, we have 



^ ^ ^r~\-^i + H \ = + ;r" + 2^ ^— r ^ . (34) 



oXn \_ at J d.v„ d/ dXn m=i OTTm oiCm aA:„ 



By (ii) and (34), we have the second set of canonical equations 



. _ dE 



TTn — i: — • 



dXn 



This completes the demonstration. 



If H does not involve the time explicitly, we can write 



W = S - Et, (35) 



where E is an arbitrary parameter, and 6* is a solution of the differential 

 equation 



H[xi, X2, xs, dS/dxi, dS/dx2, dS/dxi] = E. (36) 



The complete solution of (36) contains three arbitrary constants (besides 

 the parameter E), of which one is merely additive, and can be neglected. 

 It is easily seen that the solution of the canonical equations determined in 

 the way described above, using the function W given by (35), and treating E 

 as one of the a's, represents a motion of the particle with the total energy E. 

 All of this theory holds both for the relativistic case and for the New- 

 tonian case, the only difference being in the forms of the differential 

 equations (27) and (36) in the two cases. 



IX. Curvilinear Coordinates 



In all of the foregoing we have employed rectangular coordinates, because 

 they afford the simplest and most direct expression of the basic physical 

 facts. However, in the solution of particular problems it is often more 

 convenient to use other systems of coordinates. For this reason, we shall 

 now formulate the more important equations in terms of general curvilinear 

 coordinates. In this work, as in all work with general coordinate systems, 

 we shall encounter concepts and relations which can be handled most 

 perspicuously by means of the modern tensor calculus. Actually, the 

 amount of tensor calculus we shall use is very slight, and no extended pre- 

 liminary discussion is necessary in order to make the formulae intelligible. 

 It will suffice to give occasional explanations of the notation, and of some 

 of the concepts, as we proceed. Further information is to be found in the 

 works cited in the bibliography. 



First consider the Lagrangian equations, which, as we have seen, are 

 merely the Eulerian equations which follow from Hamilton's principle. 



