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

Abstract

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 high-frequency 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.