The Fundamental Equations of Electron Motion 

 (Dynamics of High Speed Particles) 



By L. A. MacCoU 



I. Introduction 



In work relating to the motion of electrons and other particles it is fairly 

 common to assume that the particles obey the laws of Newtonian dynamics. 

 That is, briefly, it is assumed that the rectangular coordinates (x, y, z) of 

 the particle under consideration satisfy the differential equations 



mx = X, my = Y, mz — Z, 



where m is the mass of the particle (assumed constant), X, ]', and Z are the 

 components of the applied force, and the dots indicate differentiation with 

 respect to the time t. 



However, it is well recognized now that the above equations are not 

 strictly correct, and that they merely represent an approximation which is 

 adequate when the speed of the particle is sufficiently small compared with 

 the speed of light. The system of dynamics based upon the correct equa- 

 tions^ (which will be exhibited presently) is commonly called relativistic 

 dynamics, not because any knowledge of the theory of relativity is essential 

 to its understanding and use-, but because it is in agreement with the theory 

 of relativity (which Newtonian dynamics is not), because it was first de- 

 veloped in connection with work on the theory of relativity, and because 

 even yet virtually all of the expositions of the subject are to be found in 

 books and papers dealing primarily with the theory of relativity. 



Just where the dividing line should be set between cases in which New- 

 tonian dynamics is an adequate approximation and cases in which it is 

 necessary to use relativistic dynamics is, of course, a rather vague question 

 which cannot be answered simply and definitely. We may note, however, 



1 It is not the purpose of this article to discuss questions of fundamental physics, or 

 the physical validity of any particular equations. For purposes of discussion, we assume 

 outright that relativistic dynamics is at least more nearly correct than is Newtonian 

 dynamics. 



* The theory of relativity can be described briefly as a theory of the relations between 

 the descriptions of phenomena in terms of different systems of reference. We shall not 

 be concerned with this theory, because we shall be employing the same reference system 

 throughout most of our discussion. In the final section of the paper we shall consider 

 purely geometrical transformations of the coordinate system. These transformations, 

 however, involve nothing that is really characteristic of the theory of relativity in the 

 usual sense. 



153 



