General Dynamical Principles. 581 



and since U is independent o£ i/r 1? from (18) we have 



#1 B E /9(V> 



-dt=Wi ( } 



The set of equations corresponding to (19) and (20) for all 

 the coordinates $i$ 2 tf> z ...<f) n and the momenta ^fatys ••• ty* 

 are the desired equations of motions in the canonical form. 



The expression T' + U was called E since it actually 

 turns out to be, as in the classical mechanics, the total 

 energy of the system. To show this, we have by definition, 

 equation (15), 



T / = ^ 1 +^ 2 + ... -T, 



<3</>i "dfa 



■T, 



But T by definition, equation (7), is 



— r =Swo(l~? 2 /c 2 ) *q^j-i 

 091 091 



which gives us 



. "dq . "dg 



( 1 1>1 



3y 



= 2o, n* i 3^ + * s ^ s " 



-T.i 



(21) 



We can show, however, that the term in parenthesis is 

 equal to q. If the radius vector r determines the position of 

 the particle in question, we have 



7 



*9_ 



B<j>2 



dt 

 Mi 



a/( ) 



B</>2 



. <£n) 3 

 5^1 * B</>2 



Br 



Br 



df 2 



&c, 



and hence the term in parenthesis in equation (21) become* 



{+i 





</>2 



5r 



Be/), 



+ . 





= £< 



