61, 62] HAMILTON'S PKINCIPLE. 119 



There are m of the equations (7), one for each coordinate q. 

 These are Lagrange's equations of motion in generalized co- 

 ordinates. 



62. Proof independent of Hamilton's Principle. 



We will verify these equations by direct transformation of the 

 equations in rectangular coordinates 



. 1 . 1 



which are obtained from equation (15) of Chapter IT. by means of 

 d'Alembert's principle. 



Multiplying these respectively by 



dx r dy r dz r 



dq s ' dq s ' dq s ' 



adding and summing for all values of r, the coefficient of X x be- 

 comes 



. 



r dq s dy r dq s z r d 



If there are no relations between the q's, the expression 

 <i (?!> 2m) = 



is an identity, and all its partial derivatives -^ are equal to zero. 



vqs 



Accordingly the terms in X l5 X 2 , ... disappear. 

 We have then 



T r yr Z 

 JL r = -- h I ro~" "r ^r oT~ 



I 9^8 dq s dq 



Now T= 



(12) 



