On stability of permanent rotation of a disk that collides with an horizontal plane

Stability of the motion of a thin homogeneous disk in a uniform gravitational field above a fixed horizontal plane is investigated. Collisions between the disk and the plane are assumed to be absolutely elastic, and friction is negligible. In unperturbed motion, the disk rotates at a constant angular velocity about its vertical diameter, and its center of gravity makes periodic oscillations along a fixed vertical as a result of collisions. The stability problem depends on two dimensionless parameters characterizing the magnitude of the angular velocity of the disk and the height of his jump above the plane in the unperturbed motion. An exact solution of the problem of stability is obtained for all physically admissible values of these parameters.