36, 37] LAGRANGE'S EQUATIONS. 



we may integrate the second term by parts. Since the initial and 

 final configuration of the system is supposed given, the dq's vanish 

 at t = t and t = t^ so that the integrated part vanishes, and 



Now if all the dq's are arbitrary, the integral vanishes only if 

 the coefficient of every dq s is equal to zero. Therefore we must have 



45) d(T-W) _d_t<) (T-W} 



dq s di ( dq' g - 



or if we write L for the Lagrangian function T W, 



46) (**L\-?L. 



' dt\d^)~dq s 



Since the potential energy depends only on the coordinates 





j- = 0, and we may write the equation 45) 



There are m of these equations, one for each q. These are Lagrange's 

 equations of motion in generalized coordinates, generally referred to 

 by German writers as Lagrange's equations in the second form. 

 Their discovery constitutes one of the principal improvements in 

 dynamical methods and we shall refer to them simply as Lagrange's 

 equations. 1 ) 



If the system is not conservative, by 34, 4) we must write 



fl ft 



48) l($4 8A)dt = l(dT 



J J 



from which we easily obtain 47), except that P s is not now derived 

 from an energy function. 



37. Lagrauge's Equations by direct Transformation. 

 Various Reactions. On account of the very great importance of 

 Lagrange's equations, it is advantageous to consider them carefully, 

 from as many points of view as possible. The deduction from 

 Hamilton's principle is one of the simplest, but does not perhaps 

 appeal as strongly to our physical sense as is desirable. Of course 

 as Hamilton's principle is completely equivalent to d'Alembert's, and 

 that to the equations of motion of Newton, we might have derived 



1) Lagrange, Mecanique Analytique, Tom. I, p. 334. 



