Relations in General Dynamics. 35 



This form of the theorem is given by Routh (Adv. Rigid 

 Dynamics, § 479) but is obtained by a different process. 

 The mode of derivation of the theorem (in the ordinary and 

 in the extended form) from the Hamiltonian function seems 

 to be the simplest possible. 



16. If besides forces due to a function V which depends 

 on the coordinates q lf ... ., q k of the system considered, there 

 are forces, Q, independent of these coordinates (it may be 

 which are functions of the unknown eonfigurational co- 

 ordinates of another system) bnt involve, perhaps, the time, 

 the typical Lagrangian equation for such a case is 



^BT_BT___dV 



dfbq -dq' -dq + ^ ' ' ' ' W 



where T is a function of t, the q's, and the q's. If, as in 

 what goes before, H denote %(pq)—T + Y, and be expressed 

 in terms of t, the <^s, and the j>'s, it corresponds to the two 

 equations 



dp _ dH 



dq_~bH ( 

 dt-~dp> ' " ' ^ 



or if H / = H-S(Q^) 





dp _ dH' 

 dt ~dq ' 



dq_-bW 



dt ~ ^ • ' * ' ^ 



The part H' — H is of course here supposed left in the form 

 — 2(Qg r ). The Hamiltonian differential equation for this 



case is 





^t 



= 0, (4) 



where H' is expressed in terms of q 1} q 2 , ... ., q k , t and 

 constants. As before the values of the p's are given by 

 equations of the type 



'-H (5) 



It is interesting to notice that Hamilton's principle and 

 Lagrange's variational equation and therefore also the 

 reciprocal theorems given above, remain unaffected by the 

 change in S. We take S x as given by the equation 



S^P {2(pq)-H. + %(Qq)}dt. . . . (0) 



D2 



