Relations in General Dynamics. 



41 



If, however, the <j> relations involve t, that is are of the 

 form (£(?i', q-2, • • • •, ft', ?i, ?i> • • ■ - ft> *) = °> we have 



3<fc 



3*i 



4^ + 2^+^=0 

 3<? 3^ o^ 



V 3</>A 



s §7 



3 9^ 



3^ 



dq' + Z~rdq+- -dt = 



3<ft& 

 B* 



(6) 



For p.', /> we get again equations (4), that i: 



B^ _ BW 



ft 



>-s^+sx 



■B : 





(7) 



B?/ '—d?/' JJ By, 



but instead of (5) 



Sp'^'-^ 3 =rfW-(^+2\|f)*,. . (8) 



where "W is now an exact differential of the variables 



This equation may be written, since 



%pdq — Hdt==d8, 

 in the form 



Vrf ? '-(H-^-2^)*=rf(W + S). . (9) 

 From this it follows that, if we write 



H'=H-^-SA|f : .... (10) 



the equations of motion in terms of the new coordinates are 

 of the form 



§XL= -W dq' _~dW 



dt ~dq' ' dt "3/ ' * ' 



(11) 



On account of the (^-relations the partial differential 

 equation (9) has not the exact Hamiltonian form, which, 

 however, can he supplied by a substitution. Remembering 

 that each <j> is zero we get, if we write U=W + 2\<£, 



, BW _ B<£ dU 



H'~b<?/ 



A =^ + 2xM=^ ; (12) 



d</j B<7 ; dq/ 



bu_bw. ,^B» 



& ~ b< + -*-ar 



(13) 



