EQUATIONS OF ELECTRON MOTION 161 



Newtonian dynamics, the first and second terms being the kinetic energy 

 and the potential energ}^, respectively. The equation itself is called the 

 energy integral. Similarly, we call (15) the energy integral in relativistic 

 dynamics, and we call the expression 



2r^ 2 -21-1/2 I Tr 



the relativistic energy. This energy is the sum of three parts: the proper 

 energy nioc'^, the relativistic kinetic energy 



2t. 2 -2i-l/2 2 



moc [I — V c \ — ntoc , 



and the potential energy V. 



The totality of possible trajectories of a particle in a static field of force 

 forms a five-parameter family. We now see that if the field of force is 

 static and of the kind we are considering now, the five-parameter family of 

 curves consists of ^ ^ four-parameter subfamilies, each of which corresponds 

 to a different value of the energy of the particle. Each of these four-param- 

 eter subfamilies is called a natural family of trajectories. We proceed 

 to derive the differential equations defining a natural family. 



If the constant in the right-hand member of equation (15) is denoted by 

 the symbol E, we have the relation 



ii[i + A-r + x7r - c[i - mic^iE - vrr\ m 



where 



Hence, 



.r2 = dx-z/dxi, X3 = dx^/dxi 



dt = cr-\\ + .v^ + .r;Y''[l - nilc' (E - VrV dx,. 

 From this, and the two equations 

 d moX2 dV 



dt{\ - vH-^y- dx2 



. VdAs _ dAfl _ . VdA^ _ dAil 



* ^ |_ dx2 dxs J [_ dxi dX2 J ' 



moxs dV ^ rdAi _ dA/] _ ^ VdAs _ dAfi 



\_ dxs dxi J |_ dx2 dx3 J ' 



dt (1 - z;2c-2)i/2 ar 



it follows that we have the following system of differential equations defin- 

 ing the natural family of trajectories corresponding to the total energy E: 



' In the theory of differential equations, an equation relating the unknowns involved 

 in a system of differential equations, their derivatives of orders less than the highest orders 

 appearing in the system, the independent variable, and one or more arbitrary constants, 

 is called an integral of the system of differential equations. 



