Mr Darwin, Lagrangian Methods for High Speed Motion 57 



2. We first consider the motion of a single electron in an 

 arbitrary electric and magnetic field varying in any manner with 

 the time and position. If m is the mass for low velocities, the 

 momentum is known to be mv/^, where ^ = V 1 — v^jc^. Starting 

 from this we have quasi-Newtonian equations of motion of the 

 type 



lir*}-^^ '^■^'- 



The force F^. is given from the field E, H as the vector eE + ^ [v, H], 



where v is the velocity vector of the particle's motion. E and H 



can be expressed in terms of the scalar and vector potentials in 



1 3A 

 the form E = — grad (t> ~ ^.^^ and H = curl A. 



C ct 



Then if r^ is the vector x, y, z we have as the vector equation of 

 motion 



It \t '4 ^~'' ^''^ '^ ~ c W + c ^'1' '""'^ ^^ •••(^■^^' 



where ^^ = V 1 - V/C^. 



Let q be any one of three generalized coordinates representing 

 the position of the particle. Take the scalar product of (2-2) by 



i=r^. Then since ^ = -^, we have 

 oq cq cq 



dii d (nil . )\ d (m 



dq'dtX^^'^^U dt\^. 







dt dq dq 



'9r _ . ,\_ d4> 



where "Wq = j;^ — ^ ^^^ Lagrangian operator. 



Again - gj [J-, grad .^ j = - e^ ^ = e^Bc/,. 



The remainder can be reduced to 



CV''~dq)~c[dq'lt) ^-'^^' 



dA dA dA . dA . 8A . 



where -j7 = ^7 + ^^+^-y+'^^ 



dt dt ox Cy '^ oz 



