First Integrals and Symmetries of Nonholonomic Systems

In nonholonomic mechanics, the presence of constraints in the velocities breaks the well-understood link between symmetries and first integrals of holonomic systems, expressed by Noether’s Theorem. However, there is a known special class of first integrals of nonholonomic systems generated by vector fields tangential to the group orbits, called horizontal gauge momenta, that suggests that some version of this link still holds. In this paper we prove that, under certain conditions on the symmetry group and the system, the (nonholonomic) momentum map is conserved along the nonholonomic dynamics, thus extending Noether’s Theorem to the nonholonomic framework. Our analysis leads to a constructive method, with fundamental consequences to the integrability of some nonholonomic systems as well as their hamiltonization. We apply our results to three paradigmatic examples: the snakeboard, a solid of revolution rolling without sliding on a plane, and a heavy homogeneous ball that rolls without sliding inside a convex surface of revolution.