35] PRINCIPLE OF LEAST ACTION. 101 



so that instead of 2) we have 



Although the times of arriving at corresponding configurations are 

 not the same, so that t Q and ^ are not the same as before, the 

 terminal positions are still given, so that the integrated parts still 

 vanish. Now introducing our new assumption, of variation according 

 to the equation of energy, we obtain 



7) = 12 (dTdt -f TdSf), 



to 



that is, 



8) dCzTdt = 0. 



^o 



/' 



The integral A = 1 2Tdt, which is twice the mean kinetic 



to 



energy for equal intervals of time multiplied by the time occupied 

 in the motion, is called the Action. 



Accordingly the principle stated in equation 8) is known as the 

 Principle of Least Action. 



The definition of action is usually given otherwise, for since 



ds 



9) 



A == J2Tdt =^r jm r v r ds r , 



which exhibits the action as a sum of line integrals of the momentum 

 of the particles. We may thus define the action as the sum for all 

 the particles of the mean momentum for equal distances multiplied 

 by the distance traversed by each particle. 



In the enumeration of the conditions there is now a difference 

 - the initial and final configurations of the system (positions of all 

 the points) are given as before, but instead of prescribing the dura- 

 tion of the motion, t t Q , we prescribe the initial energy k. Thus 

 in the variation of the paths the energy is supposed to be unchanged. 

 In forming the integral i is supposed to be eliminated and all the 



