98 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OE MOTION. 

 If there is a force -function U we have 



Z(Xdx + Ydy + Zdz) = dU, 



consequently the right-hand member of 1) is 



dT+dU. 



The left-hand member being an exact derivative we may inte- 

 grate with respect to t, between any two instants # and t l9 



* 

 / 



U) dt = d(T + U) at. 



t 



If the positions are given for t Q and t , that is if the variations 

 dx, dy, z vanish for t Q and t lf then the integrated parts vanish, and 



or 



3) 8 C(T-W)dt = 0. 



This is known as Hamilton's Principle. 1 ) It may be stated by 

 saying ihat if the configuration of the system is given at two 

 instants t Q and t 1} then the value of the time -integral of T -\- U is 

 stationary (that is less or greater) for the paths actually described in 

 the natural motion than in any other 2 ) infinitely near motion having 

 the same terminal configurations. 



Considering the signification of a definite integral as a mean 3 ) 

 we may state equation 3) in words as follows: The time mean of 



1) Hamilton. On a General Method in Dynamics. Phil. Trans. 1834. 



2) It is understood that both the natural and the varied paths are smooth 

 curves, that is without sharp corners. 



3) The arithmetical mean of a number of quantities is defined as their 

 sum divided by their number. A definite integral is defined as the limit of a 

 sum of a number of quantities as their number increases indefinitely. If we 

 divide the interval ab into n parts of length d g and if we denote by f g the 

 value of a function f(x) when x lies at some point within the interval ^ we 

 define 



b 



r 



l 



/ 



f(x)dx as lim 



