A Top on a Vibrating Base: New Integrable Problem of Nonholonomic Mechanics

A spherical rigid body rolling without sliding on a horizontal support is considered. The body is axially symmetric but unbalanced (tippe top). The support performs highfrequency oscillations with small amplitude. To implement the standard averaging procedure, we present equations of motion in quasi-coordinates in Hamiltonian form with additional terms of nonholonomicity [16] and introduce a new fast time variable. The averaged system is similar to the initial one with an additional term, known as vibrational potential [8, 9, 18]. This term depends on the single variable — the nutation angle $\theta$, and according to the work of Chaplygin [5], the averaged system is integrable. Some examples exhibit the influence of vibrations on the dynamics.