Dynamical Motions of Charged Particles. 545 



corresponding to each component of r l5 it is easy to see that, 

 extending the use o£ the vector notation, we have 



H =2 fi- - s al-i + Xtf -tjP- Oh, A) + SS^ 2 



(Pi> P2) , (Pi, r 2 -r0(p 2 , ro-rO") 



2Lrm l w 2 (. r 12 



,, 3 

 '12 



All the developments o£ general dynamics (such as the 

 Hamilton- Jacobi partial differential equation etc.) follow at 

 once, with the exception of such theorems as depend on the 

 kinetic energy having a quadratic form. 



For many problems it will be quicker to work in the 

 Lagrangian form direct. When q s does not occur explicitly 



in L, we have an integral ^- = p s a constant, and when 



this coordinate is " ignored " the modified Lagrangian is 

 L'=L— pj} 8 . The energy integral will exist when the ex- 

 ternal fields <j) and A do not contain the time explicitly and 



is then of the usual form 'E</ S ^r- L = const. Applying 



oq s rr J * 



this to (11) we have the integral 

 t^m^ + Xim^ + te^Xt^ 2 



2(J' 2 1 r r 3 j 



= const. . . . (13) 



The first two terms can be obtained either direct in the 

 expanded form, or else from the fact that 



*»k ( --* )+ *=wA 1 -v)~ 1 2 ' 



agreeing with the known fact that the kinetic energy of an 

 electron is mC 2 /{3. 



6. We now apply these results to the " Problem of Two 

 Bodies.'' Take e 1 =—e, m 1 = m and ^ 2 = E, m 2 = M. The 

 motion is supposed to take place in a plane and the particles 



