General Dynamical Principles. 579 



But £0!= --(^j), which gives us 



Since, however, at times «! and t s , Scpi is zero, the first term 

 disappears and we may write for (12), 



Since, however, the limits £ x and £ 2 are entirely at our 

 disposal, we must have at every instant 



Finnlly, if the </>'s are independent parameters as assumed, 

 we can assign perfectly arbitrary values to 8</> l5 &/> 2 , &c, and 

 hence must have the series of equations 



^/BL\ 3L _ - 



Ma.fr/ 3*i ' | 



rf/BL\ aL _ L ... ( 13) 



^v^>; a* 2 - Uj 



&c. 



These correspond to Lagrange's equations in ordinary 

 mechanics, differing only in the definition of L. 



Equations of Motion in the Hamiltonian I 



arm. 



We may now derive the equations of motion in the Hamil- 

 tonian or canonical form. 



Let us define the generalized momentum -^ corresponding 

 to one of the coordinates <fr by the equation 



BT 



*-*£ <"> 



It should be noted that the generalized momentum is not 

 as in ordinary mechanics the derivative of the kinetic 

 energy with respect to the generalized velocity, but ap- 

 proaches that value at low velocities. 



2P2 



