On the dynamics of a heavy symmetric ball that rolls without sliding on a uniformly rotating surface of revolution

Abstract

We study the class of nonholonomic mechanical systems formed by a heavy symmetric ball that rolls without sliding on a surface of revolution, which is either at rest or rotates about its (vertical) figure axis with uniform angular velocity. The first studies of these systems go back over a century, but a comprehensive understanding of their dynamics is still missing. The system has an $SO(3) \times SO(2)$ symmetry and reduces to four dimensions. We extend in various directions, particularly from the case $Ω= 0$ to the case $Ω\ne 0$, a number of previous results and give new results. In particular, we prove that the reduced system is Hamiltonizable even if $Ω\ne 0$ and, exploiting the recently introduced `moving energy', we give sufficient conditions on the profile of the surface that ensure the periodicity of the reduced dynamics and hence the quasi-periodicity of the unreduced dynamics on tori of dimension up to three. Furthermore, we determine all the equilibria of the reduced system, which are classified in three distinct families, and determine their stability properties. In addition to this, we give a new form of the equations of motion of nonholonomic systems in quasi-velocities which, at variance from the well known Hamel equations, use any set of quasi-velocities and explicitly contain the reaction forces.