114 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



In virtue of the homogeneity of T as a function of the q"s, we 

 have by Euler's theorem for homogeneous functions, 



38) 



r=l 



a property of which frequent use will be made. 



The potential energy, if the system is conservative, on the other 

 hand, depends only on the configuration of the system, that is on 

 the coordinates #, the q"s not appearing. For instance in the 

 problem of 23, 



W = mgz = mgr cos #. 



Whether the system is conservative or not the element of work 



r=n 



39) dA.= y>(Xrdx r + Y r dy r + Z r de r ) 



is a homogeneous linear function in the dq's which we will write 



40) d A = P 4 dq, + P 2 dq 2 + + P m dq m . 



By analogy with rectangular coordinates we shall call P r x the 

 generalized force -component corresponding to the coordinate q r and 

 velocity ql. 



If the system is conservative, since 



41) dW=-dA, P r =- 



and in any case 



dXr + Y 8Vr 4 7 r 



r Tr + Zr 



r=l 



We may now make use of Hamilton's Principle to deduce the 

 equations of motion in terms of the generalized coordinates q. 



Performing the operation of variation upon the integral occurring 

 in Hamilton's Principle, both the g's and #"s being varied, we obtain 



to 



and since 



dq 



