EQUATIONS OF ELECTRON MOTION 167 



Also, it can be shown without difficulty that in order that we have (21), for 

 all such choices of v?, xp, w, it is necessary that .v, y, and z, satisfy the equations 

 (20).* 



The last result can be stated in the following summary, and not quite 

 explicit, form: If, and only if, the functions x(/), y(t), z(/) satisfy equations 

 (20), the integral 



f 



J T 



F{t, X, y, z, x', y', z') dt (22) 



is stationary with respect to infinitesimal variations of the functions ;v(/), 

 y(/), z{t) which leave the terminal values unaltered. 



The problem of finding functions which render the values of definite 

 integrals stationary is the chief subject of the Calculus of Variations. 



The equations (20) are called the Eulerian equations of the Calculus of 

 Variations problem of making the value of the integral (22) stationary, or, 

 as we usually say, of maximizing or minimizing the integral. 



VII. Hamilton's Principle and the Principle of Least Action 



We immediately recognize equations (6) as the Eulerian equations of a 

 problem in the Calculus of Variations. Thus we have the following principle 

 {Hamilton'' s principle): 



The particle moves, under forces of the type (4), so that the value of the integral 



Ldt, 



/ 



Jit 



with /i and ti held fixed, is stationary with respect to infinitesimal variations of 

 the functions Xn{t) which leave the initial and final points unaltered. 



The precise meaning of this is determined by the discussion given in 

 Section VI. 



Hamilton's principle leads to the relativistic or Newtonian differential 

 equations of motion, according as we use in it the function L given by (5) 

 or by (5'). 



A little inspection suffices to show that the system of equations (17) is 

 also the system of Eulerian equations of a problem in the Calculus of 



d dp dp 



* In brief, suppose that — — were not zero for some value of /. Then if we should 



at dx dx 



choose a function <p(/) which was (say) positive in the neighborhood of that value, and zero 

 elsewhere, the integral 



r^2 fdP _ d_dF^l 



J J, \dx dt dx' \ 



dt 



would have a value other than zero. We shall not give the actual proof here; it is to be 

 found in the works on the Calculus of Variations cited in the bibliography. 



