342 GENERALIZED COORDINATES 



Lagrange's Equations ~by Direct Transformation 



275. Instead of deducing Lagrange's equations from equation 

 (156), they may be obtained directly by transformation of the 

 equations of motion. 



We have, as before, 



so that, on differentiation, 



dx^ df_ d6 df_d0s 



dt ~ 0^ dt 86 2 dt ' 



*-'' + '+ -' < 178 > 



Thus x is a linear function of V 2 , , and 



- - ' 



^ ~ ^ 



We have T = m (x* + f + z 2 ), 



(179) 



so that T, as before, is a quadratic function of 6^ 2 , , which 

 also involves lt 2) - -. By differentiation, 



dT /. dx 



dx . dy . dz\ 

 ~r + y -7- + * -T- I 



a^ ^ x a^/ 



or, by equation (179), 



Thus 



. . d 



+ + (180) 



