Maxwell strata and symmetries in the problem of optimal rolling of a sphere over a plane

The problem of rolling of a sphere over a plane without slipping or twisting is considered. It is required to roll the sphere from one contact configuration into another one so that the curve traced by the contact point be of minimum length. Extremal trajectories in this problem were described by Arthur, Walsh and Jurdjevic.
In this work, discrete and continuous symmetries of the problem are constructed and fixed points of the action of these symmetries in the inverse image and image of the exponential map are studied. This analysis is used to derive necessary conditions for optimality; namely, upper bounds on the cut time along the extremal trajectories.
Bibliography: 21 titles.