On some aspects of the dynamics of a ball in a rotating surface of revolution and of the kasamawashi art

TL;DR

This paper analyzes the dynamics of a rolling ball on a rotating surface of revolution, exploring stability and asymptotic behaviors, with applications to Japanese kasamawashi art, using symmetry reduction techniques.

Contribution

It introduces a symmetry-based reduction approach to study the nonholonomic system and characterizes motion behaviors and stability in different symmetric configurations.

Findings

Existence of motions asymptotic to the vertex on symmetric surfaces

Stability analysis of equilibria in inclined surface configurations

No blow-up phenomena in the studied system

Abstract

We study some aspects of the dynamics of the nonholonomic system formed by a heavy homogeneous ball constrained to roll without sliding on a steadily rotating surface of revolution. First, in the case in which the figure axis of the surface is vertical (and hence the system is $\textrm{SO(3)}\times\textrm{SO(2)}$-symmetric) and the surface has a (nondegenerate) maximum at its vertex, we show the existence of motions asymptotic to the vertex and rule out the possibility of blow up. This is done passing to the 5-dimensional $\textrm{SO(3)}$-reduced system. The $\textrm{SO(3)}$-symmetry persists when the figure axis of the surface is inclined with respect to the vertical -- and the system can be viewed as a simple model for the Japanese kasamawashi (turning umbrella) performance art -- and in that case we study the (stability of the) equilibria of the 5-dimensional reduced system.