Relations in General Dynamics. 39 



converted into a function of: the q' 's and the #'s. Equation 

 { 1) then gives 



, aw aw m 



and there are 2 k such equations. 



But by the Hamiltonian differential equation 



tpdq-Bdt=dS, (3) 



where JS is a complete differential of a function of the q's, 

 t, and h constants (the a's say), we have 



Hence, if we add (1) and (3) together we obtain 



%p'dg'-Hdt = d{W + S) (5) 



Since -p + BS/S?, that is 3(W + S)/S?, =0, W + S does not 

 contain g, and is now an exact differential of the variables 

 (q' } t), so that 



,_ a(w+s) TT_as_ B(w+s) 

 ^~ -dq' ' a* ~ " a* * ■ w 



We infer that if H be expressed as a function of the 

 variables ( //, </, f), we shall have the equations of motion in 

 the canonical form 



d£_ _dH ,Y_aH 



at b?" dt -"dp 1 ( '^ 



The permanence of the canonical form of the equations of 

 motion in any contact transformation of variables, and the 

 identity of the function H for both sets of variables are 

 thus established. The Hamiltonian principal function 

 S'iq', a f , i) [not to be confounded with the second Hamil- 

 tonian function referred to several times above] is given by 

 the equation 



®{q',a',t)=W(q>,q) + S(q,a,t). ... (8) 



The h constants a' may be regarded as corresponding to the 

 a's as the q' J s correspond to the ^'s. The partial differential 

 equation is 



^ +H W ',„V) = (9) 



