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







Now differentiating 38) totally 



dT dq' r 

 Subtracting equation 68) from 69) the terms ~ - f -j cancel and we 



hav eft 



dT f f d SdT\ dT dq r 



But this exactly the left-hand member of the equation of activity 67). 

 Thus if the system is conservative, since 



d ^ - _ dw * - dw 

 dt dt dt dt 



so that the equation of conservation of energy is always an integral 

 of Lagrange's equations. 



39. Hamilton's Canonical Equations. Although the equa- 

 tions of Lagrange are by all odds those most frequently used in 

 dynamical problems, yet in many theoretical investigations a trans- 

 formation introduced by Hamilton is of importance. 



The kinetic energy being a quadratic form in the velocities q f 

 [equation 35)], the momenta p r being the derivatives of T by the 

 q"s are, as we have seen, linear forms in the q^s. 

 dT 



53) 



n n* _L n /' _i_ _i_ n ^ 



Pm = Q- / = = Vml #1 T Vm2^2 ~r ' ' ' ~T ^mrnqm- 

 OC Lm 



These linear equations may be solved for the g^'s, obtaining any 

 as a linear function of the jp r 's, say, 



n i \ . l ~D ., [ ~D ^ i i ~D 



the J?'s being minors of the determinant, 



I Qml) Qm2> - Qmm i 



divided by D itself. 



The R's accordingly , like the 's, are functions of only the 

 coordinates q. Maxwell calls them coefficients of mobility. The 

 solution of the equations assumes that the determinant D does not 

 vanish. This is always the case, being one of the conditions that T 

 is an essentially positive function. 



