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

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 $\text{SO(3)}\times \text{SO(2)}$-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 $\text{SO(3)}$-reduced system. The $\text{SO(3)}$-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.

References in corpus (4)

• Conservation of energy and momenta in nonholonomic systems with affine constraints
• Conservation of `moving' energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces
• Moving energies as first integrals of nonholonomic systems with affine constraints
• On the dynamics of a heavy symmetric ball that rolls without sliding on a uniformly rotating surface of revolution