36] GENERALIZED COORDINATES. 



d *- V d & - V 



dt ^ dt ~~ ^ 



we find for the kinetic energy, 



29) T = -i. m (Etf + 2Fq^ + Gqft. 



This is a typical example of the employment of the generalised 

 coordinates introduced by Lagrange, q l and q 2 being the coordinates, 

 #1, q 2 the velocities corresponding, and T being a homogeneous 

 quadratic function or quadratic form in the velocities q[, q 2 , the 

 coefficients of the squares and products of the velocities being 

 functions of the coordinates alone. We shall show that this is a 

 characteristic property of the kinetic energy for any system depending 

 upon any number of variables. 



In the case of a single free particle we may express the coor- 

 dinates x, y, 8, in terms of three parameters q lf q 2 , q 3 , and we shall 

 then have as in 25) and 26) 



30) ds 2 = E u dq 2 + .E 22 dq<? + E^ dq s 2 + 2^ 12 dq l dq 2 



+ 2E 25 dq 2 dq s + 2E sl dq s dq,, 

 where 



o n W cx dx 3y dy . dz dz 



^ rs - dq r dq s + d<l r dq s ^ dq r dq s ' 



Thus the kinetic energy has the same property as before. 



Proceeding now to the general case of any number of particles, 

 whether constrained or not, let us express all the coordinates as 

 functions of m independent parameters, q lf q%, . . . q m , the generalized 

 coordinates of the system, 



Dinner entiating we have 



dx dx doc r 



32) 



x 

 The derivatives -^9 are all functions of all the ^'s. Squaring 



dQ. 

 and adding we obtain 



