Geometry and dynamics of affine nonholonomic rolling problems

This thesis is concerned with nonholonomic mechanical systems with affine constraints in the velocities. We introduce a class of examples which provide an affine generalization of the nonholonomic problem of a convex body rolling without slipping on the plane. We investigate dynamical aspects of the system such as existence of first integrals, smooth invariant measure and integrability, giving special attention to the cases in which the convex body is a dynamically balanced sphere or a body of revolution. We provide a framework which allows us to develop a rigorous modeling of the ANAIS billiard [50] and its generalizations. The framework concerns a class of hybrid systems (in which the constraints are piecewise smooth in the velocities) and allows us to give a proof of the phenomenon observed in [50] based on general results on existence of first integrals, reversibility and symmetry of affine nonholonomic systems. We prove that analogous phenomena occur in other examples.