Rolling Ball Problems

A spherical ball of radius delta rests on an oriented surface S embedded in R-3 and has a positive orthonormal frame attached to it. The states of the ball are the elements of the 5-dimensional manifold M = S x SO(3) and a move is a smooth path on M corresponding to a rolling of the ball on S without slipping. The moves without slipping or twisting along geodesics are called pure moves. Rolling ball problems on S are mainly related to the search of N(S), the minimum number of moves (or moves without twisting, or pure moves) sufficient to reach continuously any final state starting at a given initial state. We mention some results and conjectures relative to the case of a unitary ball (delta = 1) rolling on surfaces of revolution; important cases are: plane, sphere, cylinder and surfaces parallel to Delaunay. The dynamics giving the moves without slipping of the rolling ball problems are non-holonomic, preserve a volume and lead, in certain cases, to the existence of minimal surfaces immersed in M.