42 Canonical Relations in General Dynamics. 



Hence (9) becomes 



Z I JJq'-(K-'^)jt = d($ + TJ). . . (14) 



But. since 



c)L BS / 1K . 



pi =~wr^' (0) 



we have 



a^f5)' (16) 



that is S-f U is a function of (</', t). Equation (14) is thus 

 of the Hamiltonian form. 

 Writing as before 



H'=H-^, (17) 



we may infer that the equations of motion for the trans- 

 formed system are of the form given in (11). S + U is the 

 principal function. 



21. Let there be no ^-relations, but let the equations 

 connecting {p' 3 q) with (p, q) involve the time t explicitly. 

 Then Xp'dq* — Xpdq may be an exact differential of a function 

 of the variables ( p, q) only, when t is not varied. Hence 

 when t is kept constant 



Xp'dq'-tpdq^dW-^dt, . . . (1) 



where dW is an exact differential of a function of the 

 variables (p, q, t). When t is allowed to vary this equation 

 becomes 



Xp'dq'-Spdq=dW + mt 3 .... (2) 



where U is also a function of (p 3 q, t). 



Let now, by expression of the p'a in terms of (q' } q, t), 



W and L~ be transformed to functions of (q , q, t). It is 

 manifest that — U then becomes the partial differential 

 coefficient with respect to t of the transformed W 3 that is we 

 ha ye now 



%p l dq'-Spdq+^-dt=dW, . . . (3) 



with 



,_3W aw n __3w 



