116 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. 



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

 grate with respect to t, 



Jt 



t 



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

 So;, By, Bz vanish for t and ti, then the integrated parts vanish, 

 and 



or 

 (3) 



This is known as Hamilton s Principle *. It may be stated by 

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

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

 less (or greater) for the paths actually described in the natural 

 motion than in any other infinitely near motion. 



Hamilton's principle is broader than the principle of energy, 

 inasmuch as U may contain the time as well as the coordinates. 

 It is true even for non-conservative systems (where a force- 

 function U does not exist), if we write instead of 8 U 



61. Lagrange's Generalized Equations. By means of 

 Hamilton's Principle we may deduce the generalized equations of 

 motion. 



Suppose that by means of the equations of condition, if 

 there are any, we express all the coordinates as functions of 

 m = 3n k parameters q lt q. 2 , ... q m , which are known as the 

 generalized coordinates of the system, 



as l = a; 1 (q lt 0a,... q m ) 



Then TF, if the system is conservative, becomes a function of 

 the parameters q. 



* Hamilton. On a General Method in Dynamics. Phil. Trans. 1834. 



