134 IY. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



Thus the integral 

 94) S 



is supposed to be expressed in terms of these 2m + 1 variables. 

 Now if the initial and final coordinates are varied without varying 

 the time of transit t t Q (t the upper limit of the integral) we have 



We have however proved that under these conditions we have 



Since these expressions must be equal for arbitrary variations of 

 the #'s and #'s we must have 



96) M = M^ 3S = 



We may now, if we please, regard the initial coordinates 

 #i 7 . qm, and the initial momenta, p^ 9 . . . p y as 2m arbitrary 

 constants replacing the c , C 2 , . . . c^ m of equations 91) and 93). Then 

 the equations 97) will be the general integrals of the equations of 

 motion, for if the form of the function S is known in terms of 

 tj &, . q m , #1, - . . q m , the equations 97) are m equations involving 

 q ly . . . q m without their derivatives, which may be solved to obtain 

 the g's as functions of t and 2m arbitrary constants q^ 9 . . . g> J^ . . .pH, 

 as in equations 91). 



It has appeared as if in order to find S it were necessary to 

 integrate the equations of motion, so to obtain T W as a function 

 of the time, which being integrated would give S. If this were so 

 the statement just made would be of little interest. But this is not 

 necessary, for Hamilton showed that the function S, which he called 

 the Principal Function, satisfies a certain partial differential equation, 

 a solution of which being obtained, the whole problem is solved. 



The function S is a function of the variables g, the constants (f 

 and the time t, which thus occurs explicitly and implicitly. Differen- 

 tiating by t we have therefore 



9 %-% 



. 



Differentiating 94) by t, the upper limit, gives however 



d Jl=T-W. 



at 



