392.] CONSTRAINED SYSTEM. 221 



the k conditions (i), i.e. to introduce $n k = m new variables 

 or generalized co-ordinates, g lt q^ q m in the place of the 3 

 Cartesian co-ordinates x^ y v z v x^ z n , selecting the new co- 

 ordinates so as to satisfy the conditions (i) identically (comp. 

 Arts. 210-216). 



392. The Cartesian co-ordinates x, y, z of any one of the n 

 points, as well as their time derivatives x> y, z, are functions of 

 ,<7i> q^ q m and of the time t. We have therefore 



dx . dx , dx , dx 



with similar expressions for y and z. Thus x, y, z are repre- 

 sented as functions of the independent variables t, q^ q^ q m , 

 <1\> 4 2? '" <? m - 



Differentiating x partially with respect to any one, q, of the 

 quantities q lt q^ q m , we find 



dx = &x &x . &x . &x . 



dq dqdt dqdq^ dqdq^ 



_d_dx_ ,J_dx_ - , _d_ d^ . d dx . 



dt dq^dq^ dq ' fl ' 3ft dq ' *** ^ dq m dq ' 



We have therefore, 



dx_d^ dx_ ^_d_^2 ^. (n\ 



l$q~~~dt ^q ^q~ ' dt dq dq~ dt dq 



Again, differentiating (10) partially with respect to q, we have 

 dx__dx dy^_dy dz_fa / * 



TT T~> TT T~"> T~T ~ \*^/ 



dq dq dq dq dq dq 

 Let us also form the derivatives of the kinetic energy 



), (13) 



dT ^ f- d ^,'dy,-dz\ 

 viz. -=2m U^-+j// + ^ 



dq \ dq dq dqj 



which by (11) and (12) becomes 



dT f.d dx .d dy . -d dz\ /T ^ 



= ^m(x-- +y-r -^ + *^ TT ) (14) 



dq \ dt dq dt dq dt dqj 



