Relations in General Dynamics. 33 



expressed by finite equations. If the q's be connected by 

 non-integrable differential relations the system is not holo- 

 nomous, and a sequence of configurations given by varied 

 coordinates is not in general a succession that can be taken 

 by the system. We shall call a holonomous system quasi- 

 conservative it' the generalised forces are the partial deriva- 

 tives — 'dYj'dqi, . . . ., — dV/c^j. of a function V of the </s 

 and t. If t does not appear in V, or in the kinematical 

 conditions the system is truly conservative. 



13. We now consider the principle of Hamilton and the 

 analogous principle of Least Action for both quasi-conser- 

 vative and truly conservative systems. The representative 

 point P moves from P at time t to P at time t along a path 

 in the ^-dimensional space — along a trajectory in that space, 

 if A be a locus of possible positions of P at time r a 

 corresponding locus A will exist for each value of t. The 

 corresponding loci w T ill be of the same kind, for example 

 both curves or both surfaces. 



Now let the trajectory be varied from that joining two 

 corresponding points C , G to a neighbouring trajectory 

 joining two corresponding points D , D. If q lt q 2 , . . ; ., q k 

 be coordinates at time t and qi + Sqi, qz + ^q^ . . . ., q /c + $q k 

 be coordinates of the corresponding point in the varied 

 motion at the same instant, then since both paths of the 

 representative point are possible, and the times of starting 

 and arrival are the same for both, we have 



^= 2 (g^ +s (iH 



=2(p80)-2(58a) (1) 



If D coincide with G and D with C the value of SS for 

 the interval t — r is zero, This is Hamilton's principle. 

 Obviously it expresses the fact that 



t Ldt=0; (2) 



i 



ior by definition 



^ 1 =C t Ldt=C{t(p'i)-R}dt. ... (3) 



Thus S x is stationary in value with respect to neighbouring 

 paths of the representative point between the same terminal 

 positions. 



Phil. Mag. S. 6. Vol. 27. No. 157. Jan. 1914. 1) 



