328 GENERALIZED COORDINATES 



a series we have 



Thus we have L = 2T E, L f = 2T' E, 



/"* ^2 



so that S = I Ldt 



At 



= f\2T-E)dt 



-f 



J tl 



Thus if S is a maximum or a minimum, it follows that 



2Tdt 



is a maximum or a minimum. This integral is called the action 

 of the motion. We now see that of all possible series of configura- 

 tions which bring the system from one configuration to another in 

 a given time, and in such a way that the total energy has always 

 a specified constant value, that one which can be described by a 

 natural system is the one on which the action is a maximum or 

 a minimum. Since the action is in general a minimum, this prin- 

 ciple is known as the principle of least action. 



The statement of this principle was first given by Maupertius 

 (1690-1759), who did not deduce it by mathematical reasoning, but 

 believed it could be proved by theological arguments that all changes in 

 the universe must take place so as to make the action a minimum (Essai 

 deCosmologie, 1751). 



NON-CONSERVATIVE FORCES 



268. If the forces are non-conservative, we may no longer, as in 

 equation (154), replace 



by W, and consequently, instead of equation (156), we shall have 



dt = - ( 167 ) 



f 



