336 



GENEKALIZED COOKDINATES 



ILLUSTRATIVE EXAMPLES 



1. A homogeneous sphere of radius a rolls down the outer surface of a fixed 

 sphere of radius b without sliding. Find the motion. 



At any instant let the line of centers make an angle with the vertical, 

 and let the angular velocity of the rolling sphere be 0. The velocity of the 

 center of the rolling sphere is (a + 6)-0, so that 



The potential energy is 



W mg (a + b) cos 0, 

 so that L = T - W 



FIG. 154 



mg (a + b) cos 0. (a) 



The variations in and are not 

 capable of having any values we 

 please, for the velocity of the center 

 of the moving sphere is (a + b) 0, 

 and also must be a0', since the sphere 

 is rolling with angular velocity 0* 

 without sliding. Thus 



ad = (a + b) 0. (6) 



This is true at every instant of every possible motion, so that we must have, 

 on integrating with respect to the time, 



a0=;(a + &)0 + a constant, 



and hence we must suppose changes in the coordinates 0, to be connected by 

 the relation a 50 = (a + 6) 50. 



Thus Lagrange's equations are 



Eliminating X, we obtain 



Substituting from equation (a), this becomes 



( a + 6)ri/?maAl + af(m(a.+ &) 2 0) - mg (a + b) sin J = 





