224 MOTION OF A VARIABLE SYSTEM. [396. 



tern, and =&W for a general system (Art. 389), the equation 

 reduces to the simple form 



o, (21) 



in the general case, and to 



=0, or Sj[V- PV=o> (22) 



in the case of a conservative system. 



396. Hamilton's principle consists in the proposition that the 

 equation (21) or (22) holds for any displacements of the system 

 compatible with the conditions (i), provided these displace- 

 ments be zero at the times / x and t^. Assuming the existence 

 of a force-function, i.e. taking (22) as the expression of Hamil- 

 ton's principle, its meaning can be expressed as follows. If we 

 consider any two positions of the moving system, say the posi- 

 tions which it occupies at the times ^ and / 2 , the motion by 

 which the system actually passes from the former to the latter 

 position is characterized, and distinguished from any other 

 imaginable ways of passing from the former to the latter, by 

 the property that the variation of the time-integral of the differ- 

 ence between kinetic and potential energy vanishes. In other 

 words, for the acttial motion the average value of the difference 

 between kinetic and potential energy during any time is a 

 minimum. 



The chief advantage of Hamilton's principle lies in the 

 fact that it is independent of any co-ordinate system, and can 

 therefore be used as a convenient starting-point for introducing 

 the variables best adapted to the needs of the particular prob- 

 lem. 



397. A more complete discussion of Lagrange's equations of motion 

 and of Hamilton's as well as other similar principles, of dynamics, such 

 as the principle of least action, of least constraint, etc., will be found in 

 the work of E. J. ROUTH (see Art. 373) and in W. SCHELL'S Theorie der 

 Bewegung und der Krafte, Vol. II., pp. 544-571. The kinetics of the 



