124 THEORY OF NEWTO^ T IAN FORCES. [PT. I. CH. III. 



64. Hamilton's Transformation. We have seen, 61 (3), 

 that the kinetic energy is a homogeneous quadratic function of 

 the variables q representing the velocities, 



the coefficients being functions of the coordinates q. If we call 

 the reciprocal function with respect to the g"s, T, by the last 

 section this is also the kinetic energy, expressed not in terms 

 of the velocities, but of the momenta p. Any p s is a homogeneous 

 linear function of the q"s, so that solving the equations 



dT ^ . ^ r\ i 



+ (Jmqn , 



(1) 



- 8 - ' 



dqn 



for the q'*s, every q is a homogeneous linear function of the p's, 

 and T is therefore a homogeneous quadratic function of the 

 momenta p. By virtue of the two properties of the reciprocal 

 function we have for every q s ' (variable of reciprocation), and 

 every q g (not of reciprocation), 



(2) <?/ = 5 > 5~ = o~> 



dp s dq s dq s 



so that Lagrange's equations, 62 (17), are transformed to 

 dp s dT dW -n , dq g dT 



/ ft\ _* I . I TJ ry ' J. 6 



If we put HT-\- W , this is the reciprocal function to the 

 Lagrangian function 



and the equations take the nearly symmetrical form, 



dps = _dH p dqs = dH 



dt 'dqs dt dp 8 ' 



These are Hamilton's equations of motion. 



From these equations we may immediately deduce the integral 

 equation of energy. By cross-multiplication of the above equa- 

 tions, after transposing and summing for all the coordinates, we get 



dq s 



