EQUATIONS OF ELECTRON MOTION "163 



■•]■ 



\dA^ _ mH 



*yi-'ldxz dxiY 



k VdAt _ dA 



c{2mo*y^ L ^^1 dXi 



^ 



c(2wo 



the natural family of trajectories of a relativistic particle^ {of rest-mass mo) 

 moving with relativistic total energy E in the field of force derived from the 

 functions V, Ai, A 2, As is identical with the natural family of trajectories of a 

 Newtonian particle {of mass mo*) moving with Newtonian energy E* in the 

 field of force derived from the functions V*, Ai*, A2*, A3*. 

 In particular, the conditions of the theorem are satisfied if 



k = c{2moy'\ E* = c--{2mo)~'^{E- — mlc^), mo* = mo, 



V = V* = 0, .4i* = .li, A2* = A2, A3* = A3. 



Hence, we have the corollary: 



In the case of an electrified particle moving in any static magnetic field the 

 natural family of trajectories corresponding to any value of the energy given by 

 relativistic {Newtonian) dynamics is identical with the natural family of 

 trajectories corresponding to a certain other value of the energy given by New- 

 tonian {relativistic) dynamics. 



The equation 



E* = c~\2mo)~\E^ - mlc^) 



establishes a one-to-one correspondence between the physically significant 

 {E ^ moc"^ and E* ^ 0) values of the relativistic energy E and the New- 

 tonian energy E*. From this fact and the preceding corollary we get the 

 following further result: 



/;; the case of an electrified particle moving in any static magnetic field the 

 total five-parameter family of trajectories given by relativistic dynamics is 

 identical with that given by Newtonian dynamics. 



Of course, these peculiar properties of motion of an electrified particle 

 moving in a static magnetic field are explained physically by the fact that 

 the magnetic forces do no work, so that the speed of the particle, and 

 consequently also its mass, remain constant during the motion. 



VI. Some Formulae from the Calculus of Variations 



This section is devoted to the derivation of some formulae from the 

 Calculus of Variations which will be needed in the further discussion of the 



^ I.e. a particle obeying the laws of relativistic dynamics. 



