41] VARYING ACTION. 133 



expression for the variation of S is of great importance , for by 

 means of it we can obtain a method of integrating the equations of 

 motion , and obtaining the coordinates q and momenta p at any 

 time t . As we are now to consider the upper limit ^ as variable 

 it will be convenient to drop the subscript 1. 



Suppose we have integrated the differential equations of motion 

 completely so as to obtain every coordinate as a function of the 

 time t, involving 2m arbitrary constants, c ly C 2 , . . . c 2m? the number 

 necessarily introduced in integrating the m Lagrangian equations of 

 the second order or the 2m Hamiltonian equations of the first order. 

 Let the integrals be 



Differentiating these by t we obtain 



from which by equation 53) we may .find the ^>'s as functions of t, 



\jfjj p r (pr \tj C^j C% y . C2m)' 



These equations with 91) constitute 2m integral equations of the 

 system. , 



Inserting the particular value in our integral equations we have 



91') (f r = f r (t , C , C , . 



93') V Q . = cp (t c c 



We accordingly have the 4^m + 1 variables, 



connected by 2m integral equations. We may thus choose any 

 2m + 1 of them as variables in terms of which to express the 

 remaining 2m. 



For instance in the problem of shooting at a target 35 we 

 saw that the motion was completely determined by the coordinates 

 of the initial and final positions and the initial velocity. The latter 

 determined the time of transit , so that it together with the initial 

 coordinates, q^ , . . . 0m, an d the final coordinates, q lf . . . q m , may be 

 taken as independent variables in terms of which everything may 

 be expressed. 



