General Dynamical Principles. 577 



For this reason in our new system of mechanics we shall 

 use in the Lagrangian function instead of the kinetic energy 

 a quantity T defined by the equation 



T = 2m e»(l- v /l-3*/«»), • • • • (7) 



where the summation extends over all the particles of the 

 system. For slow velocities (i. e. small values of q) this 



reduces to 2— ^ » the kinetic energy, and at all velocities 



1 2 

 we have the relation 





?=^-\/±-qi 2 /c 2 = /i — -2 /n 2= m iqi> (8) 



\/ J- — t/i C 



Principle of Least Action. 



We are now in a position to derive a principle corre- 

 sponding to that of least action in ordinary mechanics. 

 Consider the path by which a dynamical system actually 

 moves from state (1) to state (2). The motion of any 

 particle in the system of mass m will be governed by the 

 equation 



*-*<"■> W 



Let us now compare the actual path by which the system 

 moves from state (1) to state (2) with a slightly displaced 

 path in which the laws of motion are not obeyed, and let the 

 displacement of the particle at the instant in question be Br. 



Let us take the inner product of both sides of equation (9) 

 with 6r ; we have 



Y.Br = ^(mq).or, 



= ^(mq.or)-»iq.^Sr, 



= ^(mq.6r)-mq.Sq, 



(mq . Sq + F . Bt)dt=d(mq . Br). 



Summing up for all the particles of the system, and 

 integrating between the limits t x and t 2 , we have 



C ■ I h 



I "(2/»q.Sq + 2P.8r)^ = 2mq.8r| " 

 J *, I tl 



Phil Mag. S. 6. Vol. 28. No. 166. Oct. 1914. 2 P 



