56 Mr Darwin, Lagrangian Methods for High Speed Motion 



Lagrangian Methods for High Speed Motion. By C. G. Darwin. 

 [Read 8 March 1920.] 



1. In the later developments of Bohr's* spectrum theory, it 

 is necessary to calculate the orbits of electrons moving \vath such 

 high velocities that there is a sensible increase of mass. The selection 

 of the orbits permitted by the quantum theory almost necessitates 

 the treatment of such problems by Hamiltonian methods. Working 

 on these lines Sommerfeldf and others have calculated with a very 

 high degree of success those spectra which involve the motion of 

 a single electron. But the application of the Hamiltonian function 

 involves a knowledge of the momentum corresponding to any 

 generalized coordinate, and in the formulation of most problems 

 the momenta are not known a priori but must be calculated from 

 the corresponding velocities. In other words the formation of the 

 Hamiltonian function must in general be preceded by that of the 

 Lagrangian. An exception occurs in precisely the problems referred 

 to above; for, the electromagnetic theory furnishes directly values 

 for the momentum and kinetic energy of a moving electron in 

 terms of its velocity, and the velocity can be eliminated between 

 them so as to obtain the Hamiltonian function. But in even slightly 

 more complicated cases this simple relation is destroyed — thus the 

 problem of a single electron in a constant magnetic field can only 

 be solved by introducing the artificial conception of rotating axes 

 • — and in general it will be necessary to follow the direct course of 

 finding the Lagrangian function in terms of the generalized velocities, 

 and then deducing from it the momenta and the Hamiltonian 

 function in the usual way. 



If more than one particle is in motion another difficulty enters. 

 For the interaction of two moving particles depends on a set of 

 retarded potentials and the effect of the retardation is readily seen 

 to be of the same order as the increase of mass with velocity. The 

 calculation of the retardation can only be carried out by expansion 

 and so the results are only approximate. This is not surprising since 

 the methods of conservative dynamics cannot apply to such effects 

 as the dissipation of energy by radiation, effects inevitably required 

 • by the electromagnetic theory, though they do not occur in actuality. 

 We can also see from the fact that these radiation terms are of 

 the order of the inverse cube of the velocity of light, that it will 

 be useless to expand beyond the inverse square. 



* N. Bolir, Kgl. Dan. Wet. SelsL, 1918. 



t A. Sommerfeld, Ann. Phys., vol. 51, p. 1, 1916. 



