Relations in General Dynamics. 37 



obtain by variation, and substitution for 8S X from (7), § 16, 



W 



8U+ V t~dt\=t8K-r8K + t(pAq)-S(bAa), 



where as before 



Aq = 8q + q8t, Aa = 8a + a8t. 



If at time t, and at time t, the configuration of the system 

 be the same for both paths, we have 



oYa+ (\ d ^dt\=t8K-T8K ; 



(5) 



and if, further, the terminal values of H be the same for 

 both paths, we obtain 



s(a+J'^*)=0 (6) 



The quantity 



A+ l l -dt dt 



» r 



is thus stationary in value for different paths of transition 

 adjacent to the actual path taken under the forces derived 

 from L. If H is constant during the transition, that is if 

 there is conservation of energy, the conclusion is that A is 

 stationary in value for different paths of transition adjacent 

 to the actual path. This is the principle of least action. 



It is to be observed that it is not assumed that the time of 

 passage is unvaried when adjacent paths are referred to in 

 the statement of the principle : the condition imposed is the 

 constancy of H. The value of A is then connected with 

 the Hamiltonian function S] by the equation 



A = S 1 + H(«— r) (7) 



18. With regard to systems which are not holonomous, 

 the use of the Hamiltonian function S shows that Hamilton's 

 principle holds without alteration of form for such systems. 

 Here the q's are connected by non-integrable differential 

 relations of the form 



«ii 8qi + »2i 8q 2 + + *ki 8q^ + a* 8t = 0, . . ( 1 ) 



of which there is a smaller number than k. Tf the 



