158 BELL SYSTEM TECHNICAL JOURNAL 



concerning the force is the following. V{xi, .T2, Xz, t) is the potential energy 

 of the electron in an electromagnetic field; that is 



V{xi, .T2, X3, /) = — ev?(a-i, .T2, Xz, t), 



where e is the absolute value of the electronic charge, and <p(xi, X2, xz, t) is 

 the scalar potential of the field. The functions An(xi, X2, X3, t) are related 

 to the components fl„(.ri, x^, Xz, t) of the vector potential of the field by the 

 equations 



An{xu ^1, Xz, t) = — ea„(xi, .T2, Xz, t). 



The terms —dAn/dt are — e times the contributions of the vector potential 

 to the components of the electric force. The quantity dAz/dx^ — dAi/dxz 

 is —eBi, where Bi is the .Ti-component of the magnetic induction; and 

 similarly for the quantities dAi/dxz — dAz/dxi and dA2/dxi — dAi/dx2.* 

 In other cases also, equations (4), which may degenerate considerably, can 

 be interpreted without difficulty. 



Now we define a function L{xi, X2, xz, Xi, X2, Xz, i) of the coordinates, the 

 components of the velocity, and the time, as follows: 



L= - moc^il - v^c-^)"^ - V -\- XiAi + 0-2^2 + XzAz. (5) 



We call this the Lagrangian function. 

 We write the equations 



carry out the indicated differentiations, and readily verify that the resulting 

 equations are identical with those obtained by substituting the expressions 

 (4) in equations (1). Hence, equations (6) are merely a form of the differ- 

 ential equations of motion. We call equations (6) the Lagrangian equations. 

 The chief importance of these equations is due to the ease with which they 

 enable us to use coordinate systems which are not rectangular. This will 

 be discussed in the final section. 



In the Newtonian case, i.e. the case in which the speed of the particle is 

 small compared with the speed of light, the Lagrangian function reduces 

 approximately to the form 



L = -moc"^ + -^ (^1^ + ^2^ + ^3") - V + XiAi -\- X2A2 + ^3 Az . (5') 



* These relations between the ^'s and the components of the vector potential, and 

 between the partial derivatives of the yl's and the components of the magnetic induction, 

 are based upon the use of the M.K.S. system of units. If we measure the electromagnetic 

 quantities in other units, certain constant proportionality factors may appear in the 

 relations. 



