39S-] CONSTRAINED SYSTEM. 223. 



This equation can be derived more directly from equation (16) 

 by considering an infinitesimal displacement of the system. If 

 7 be the change of the co-ordinate q in such a displacement, 

 the partial changes, or variations, of ;tr, y, z will be 



hence the work of the effective forces m'x, my, mz, for the 

 whole system, is 



, "dy , -dz\ 5- v (..die . .. dy . ..dz 

 +y-^ + z--} = q^m(x+y-/ r + z 

 \ dq ' dq dqj \ dq dq dq 



This is the amount by which the potential energy V= U is 

 diminished; it is, therefore, equal to (dU/dq}q. Hence the 

 first term in the right-hand member of (16) can be replaced 

 by dU/dq\ this at once giyes equation (19). 



395. From Lagrange's equations it is easy to derive Hamil- 

 ton's principle. 



Let each of the equations (18) be multiplied by the infinitesi- 

 mal displacement, or variation, 8$ ; let the equations be added, 

 multiplied by dt, and integrated from t to t^ : 



a (20) 



The first term can be transformed by partial integration ; remem- 

 bering that d(q)/dt = (dq)/dt, we have 



d d 



s 

 = ( Sg) I 



dt dq ' \dq Vii J** $2 



If now the variations Sq be so selected as to vanish both at 

 the time t l and at the time / 2 , the first term vanishes at both 

 limits. Hence equation (20) assumes the form 



As s &g+Sg = ST*nd 2g^ = S^7for a conservative sys- 

 dq dq 



