39, 39 a] VARYING CONSTRAINT. 129 



If the system is not conservative ; there may be still some forces 

 which are derivable from a potential energy function. In that case 

 the Hamiltonian function is to be formed with that energy, but we 

 must add to the right of equation 78 a) the non- conservative force 

 - F r ^\ Thus our equations become 



80) fe. - FV ^- a % 



dt " " dq r * r > dt~frir' 



The equation of activity then becomes 



H dq dH dp. dq r 



~ + ~ ~ 



or 



dt 

 if there is a dissipation function. 



iX 

 , 39 a. Varying Constraint. It may happen that the equations 



of constraint contain the time explicitly, that is 



<PI (t> x i> yu *!> x n> y*, **) = o, 



9>2 & Xl9 Vl9 *1?V" **9-y9 *n) = 0, 



82) . . . .......... 



Such a case is that of a particle constrained to move on a surface 

 which is itself in motion, say a sphere whose center moves with a 

 prescribed motion. The constraint is then said to be variable, and 

 the work done by the constraint no longer vanishes, for the surface 

 has generally a normal component in its motion, which causes the 

 reaction to do work. The variability of the constraint has an 

 important effect on the equations of motion. We can then no longer 

 determine the position of the system by means of a set of in- 

 dependent parameters, but must give not only their values, but also 

 the time. We rnay put 



83) 



from which, by the elimination of the g's, we may obtain equa- 

 tions 82). 





1) cf. 37, 60). 



, Dynamics. 



