Dynamical Motions of Charged Particles. 541 



Let q lz q 2 , q s be three generalized coordinates defining the 

 position of the particle. The components of 1*! are then 



known functions of the o's and ^-r - = ^— - for any com- 



a dq Oq 



ponent of rj and any of the q's. 



Take the scalar product of (4) by ^-* . 



Then 



/Bri d ini . \ _ jl f 3 /^r, • Y\ __ ™i /dij . \ 

 \dq ? dt A r V ~ dt { A \-dq ' V i A U'2 ' V 



and putting in the value of /3 1? this reduces to S? ( — mC 2 /3i), 



where 2)„ = ~t ^r- — ^^ the Laorangian operator. The ex- 

 dt oq oq 



pression — mC 2 /3 t has an obvious connexion with the " world 



line " of a particle in relativity theory. 



The next term in the equation is 



For the remainder we simplify by writing out one com- 

 ponent of the vector product, 



bA 



B^ 



/,,3A t BA, r . 3AA 



and the second factor is -—' _ Nj x , where —r- denotes the 



dt &■ dt 



total rate of change of A at the moving particle. Thus 

 e l (~dx l dA r , .-.A ^/. ^A\ ^/d^ <ZA\ 



(J W ' " * + [ri ' ° Url A] ) = C I* 1 ' 37) " C W' 2M 



So the equations of motion can be derived from a Lagrangian 

 L=-.m 1 C%-^+^(r 1 ,A). . . . (5) 



