128 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



The term p r dq' r = Pr^. Sq r being integrated by parts and the dq's 

 put equal to at the limits, we have 



78) 



Now since W does not depend upon the momentum p rt 



therefore the coefficients of the dp's all vanish. If the dq's are all 

 arbitrary, their coefficients must accordingly vanish so that we have 



the first equation being the equation of motion, the second defining 



q' r = j-^- These equations 78) were introduced by Hamilton and on 



account of their peculiarly simple and symmetrical form they are 

 often referred to as the canonical equations of dynamics. In practical 

 problems they are generally not more convenient than Lagrange's 

 equations. 



We may recapitulate Hamilton's method as follows: 

 Form the Hamiltonian function H, representing the total energy 

 of the system as a function of the 2m independent variables q and p, 

 the coordinates and momenta. Then the time derivative of any co- 

 ordinate q is equal to the partial derivative of H with respect to 

 the corresponding momentum p > while the time derivative of any 

 momentum is equal to minus the partial derivative of H with respect 

 to the corresponding coordinate. A direct deduction of the equations 

 of Hamilton without the use of Hamilton's Principle will be found 

 in the author's Theory of Electricity and Magnetism 64. 



The equation of activity is most simply deduced from Hamilton's 

 equations, for by cross multiplication of equations 78), after trans- 

 posing and summing for all the coordinates we get 



791 ^ SHdp. 



r + -~ = 



But this is equal to the total derivative of If by t, 



dJS_ 



dt ~~ u > 



which being integrated gives 



H = h, 



a constant. But since H = T + W, this is the equation of energy. 







