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LI. The Dynamical Motions of Charged Particles. By 0. G. 

 Darwin, M.A., Fellow and Lecturer of Christ's College, 

 Cambridge *. 



1. nPHE work of Bohr f and of Sommerfeld \ and others 

 -A has given a new importance to problems connected 

 with the orbits of an electron — in particular, to the effect on 

 the orbits of the increase of mass with velocity. The first 

 object of the present paper is to reduce the problem of the 

 motion of any number of charged particles, moving at high 

 velocities in any electric and magnetic field, to a Lagrangian 

 form, so that all the well-known theorems of general dyna- 

 mics may be made applicable. These principles are then 

 applied to an example, the problem of two bodies ; and, 

 finally, as a matter of some theoretical interest (though it 

 was never to be expected that the effect would be per- 

 .ceptible in practice), these results are applied, according to 

 Sommerfeld's quantum principle, to calculate the small in- 

 fluence on the doublets of the hydrogen spectrum, due to 

 the finiteness of mass of the nucleus of the atom. 



The application of the methods of general dynamics to 

 such problems is by no means new. Tims Sommerfeld 

 makes much use of the canonical form in the solution of 

 the orbits of a single electron, and much of Bohr's § later 

 work is carried out with the Hamilton-Jacobi partial dif- 

 ferential equation. Now the direct application to such 

 problems of the canonical equations of motion implies a 

 knowledge of the momenta corresponding to the various 

 generalized coordinates, whereas in the formulation of airj 

 problem it is the velocities which are known and not the 

 momenta. An exception occurs in the case of a single 

 particle in a fixed electric field. Here the linear momentum 

 is known to be mv/ S (1 — v 2 /C 2 ) ||, and the momentum cor- 

 responding to any other coordinate can be deduced by 

 elementary methods. But even for a single electron a 

 magnetic field upsets this rule, and in the case of several 

 free electrons it is quite impossible to obtain the momenta 

 a priori. In other words, for a general method of formula- 

 tion, the Lagrangian must be found first, before it is possible 

 to proceed to the Hamiltonian, and the Lagrangian, of course, 



* Communicated by the Author. 



t N. Bohr, Phil. Mag. vol. xxvi. pp. 1, 476, 857 (1913), vol. xxvii. 

 p. 506 (1914), vol. xxix. p. 332 (1915), vol. xxx. p. 394 (1915). 

 X A. Sommerfeld, Ann. d. Phys. vol. li. p. 1 (1916). 

 § N. Bohr, Kgl. Dan. Vtd. Selsk. 1918. 

 || v the velocity, C the velocity of light, m the mass at low velocities. 



Phil. Mag. S. 6. Vol. 39. No. 233. May 1920. 2 N 



