IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 97 



CHAPTER IV. 



PRINCIPLE OF LEAST ACTION. 

 GENERALIZED EQUATIONS OF MOTION. 



34. Hamilton's Principle. We shall now consider a principle 

 that differs from those of the last chapter in that it does not 

 immediately furnish us with an integral of the equations of motion. 

 On the other hand, like d'Alembert's principle it enables us to 

 embody the laws of motion in a simple mathematical expression 

 from which we can deduce the equations of motion, not only in the 

 simple form hitherto used employing rectangular coordinates, but also 

 in a form involving any coordinates whatsoever. This statement, 

 employing the language of the calculus of variations, permits us to 

 enunciate the principle in the convenient form that a certain integral 

 is a minimum. The so-called Principle of Least Action was first 

 propounded by Maupertuis 1 ) on the basis of certain philosophical or 

 religious arguments, quite other than those upon which it is now based. 



We shall first treat it in the form given by Hamilton. If in 

 d'Alembert's equation 



we consider dx, dy, 8s arbitrary variations consistent with the equa- 

 tions of condition, we have 



d*x ~ _ d (dx ,. \ dxdSx 



~W OX ~di\dt ox ) ~~di~dt 



d /dx ~ \ dx 



= di\di x )~~di 



~dt 

 dt\dt 



Treating each term in this manner, taking the sum, and removing 

 the sign of differentiation outside that of summation, 



1) Mem. de 1'Acad. de Paris, 1740. Also: Des lois de mouvement et de 

 repos deduites d'un principe metaphysique , Berlin, Mem. de 1'Acad. 1745, p. 286. 



WEBSTER, Dynamics. 7 



