  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.
