The optimal rolling of a sphere, with twisting but without slipping

The problem of a sphere rolling on the plane, with twisting but without slipping, is considered. It is required to roll the sphere from one configuration to another in such a way that the minimum of the action is attained. We obtain a complete parametrization of the extremal trajectories and analyse the natural symmetries of the Hamiltonian system of the Pontryagin maximum principle (rotations and reflections) and their fixed points. Based on the estimates obtained for the fixed points we prove upper estimates for the cut time, that is, the moment of time when an extremal trajectory loses optimality. We consider the problem of re-orienting the sphere in more detail; in particular, we find diffeomorphic domains in the pre-image and image of the exponential map which are used to construct the optimal synthesis.