100, 101] 



LAGRANGE'S EQUATIONS AND ROLLING. 



313 



where c t and c 2 are arbitrary constants. From this we obtain 



Ma* dr 

 * ~ Ma*~ d&' 



j f\ ' -t t 



and from equation 209) we obtain a = -=-> or -=- as a function of #, 



at d& 



so that the time is given in terms of # by a quadrature. The 

 explicit completion of the solution is too complicated to be of use 

 in investigating the motion. The equations 207) have been investigated 

 by Carvallo by a development in series, from which the properties 

 of the motion are investigated. 



1O1. Lagrange's Equations applied to Rolling. Noii- 

 iutegrable Constraints. In the attempt to apply the method of 

 Lagrange to the problem of rolling we are met with a peculiar 

 difficulty, which has been the subject of researches by Vierkant and 

 Hadamard. 1 ) We shall follow the treatment of the former of the 

 rolling of a disk. Let us characterize the position of the disk by 

 the angles ^, ^, <p, as before, and in addition by the coordinates x, y 

 with respect to a set of fixed 

 axes in the horizontal plane, of 

 the point of contact of the disk 

 and plane. These five coordinates 

 completely characterize the posi- 

 tion of the disk, but are not 

 all independent, on account of 

 the constraint of rolling. If we 

 measure ty as the angle made 

 by the vertical plane through 

 the normal to the disk with the 

 X-axis, as indicated in Fig. Ill, 

 we see that changes of ijs and # 



do not affect the coordinates of the point of contact, but that a 

 change d<p of (p causes a shifting whose components dx, dy are given 

 by the equations of rolling) x 



dx + adcp sin^ = 0, 



216") n 



dy a d <p cos ^ = 0, 



which constitute the equations of constraint. These equations differ 

 from any that we have heretofore met, in that they are not integrable, 

 that is, they are not, like the equations 8) Chapter III, derived from 

 equations obtained by putting certain functions of our five variables 



Fig. 111. 



1) Vierkant. Uber gleitende und rollende Beivegung. Monatshefte fur Math, 

 u. Physik, 1892, p. 31. 



