Dynamics of an Unbalanced Disk with a Single Nonholonomic Constraint

The problem of the rolling of a disk on a plane is considered under the assumption that there is no slipping in the direction parallel to the horizontal diameter of the disk and that the center of mass does not move in the horizontal direction. This problem is reduced to investigating a system of three first-order differential equations. It is shown that the reduced system is reversible relative to involution of codimension one and admits a two-parameter family of fixed points. The linear stability of these fixed points is analyzed. Using numerical simulation, the nonintegrability of the problem is shown. It is proved that the reduced system admits, even in the nonintegrable case, a two-parameter family of periodic solutions. A number of dynamical effects due to the existence of involution of codimension one and to the degeneracy of the fixed points of the reduced system are found.