578 Prof. R. C. Tolman on the Relativity Theory : 



Since t x and t 2 are the times when the actual and displaced 

 motions coincide, we have at these times 6V = 0j and we also 

 have q . 8q — q8q leading to the relation 



\(%mq8q + %V.8r)dt = 0. 



With the help of equation (8), however, we see that 

 %mq8q = 8T, giving us 



\ t2 (8T + XF.8v)dt=0 (10) 



If the forces XF are conservative, we may write 

 2F.Sr=-oTJ, 



where 8U is the difference between the potential energies 

 of the displaced and the actual configurations. This gives us 



si** (T-JJ)dt = 0, (11) 



A 



which is the modified principle of least action. The principle 

 evidently requires that for the actual path by which the 

 system goes from state (1) to state (2), the quantity 



J 



t \T-TJ)dt = 0, 



shall be a minimum (or maximum). 



Lagrange's Equations. 



We shall call T — U the Lagrangian function L. Let us 

 suppose now that the position of each particle of the system 

 is some function of the n independent generalized coordinates 

 (p 1 (j) 2 4 > z ■ • • <i>n, and hence that L is some function of 



</>l</>2</>3 • • • </>»> #1#2#3 •••#>» 



where for simplicity we have put 

 From equation (11) we have 



f' 2 sL .dt= f ''(is!?;- Hi+i^-4i)dt=o. . u2) 



