First Integrals and symmetries of nonholonomic systems

TL;DR

This paper extends Noether's Theorem to nonholonomic systems, showing under certain conditions that momentum maps are conserved, which aids in understanding their integrability and Hamiltonization.

Contribution

It proves the conservation of momentum maps in nonholonomic systems under specific symmetry conditions, extending classical results to this more complex setting.

Findings

Conservation of momentum maps in nonholonomic systems under certain symmetries

Identification of a new horizontal gauge momentum for the snakeboard

Implications for integrability and Hamiltonization of nonholonomic systems

Abstract

In nonholonomic mechanics, the presence of constraints in the velocities breaks the well-under\-stood link between symmetries and first integrals of holonomic systems, expressed in Noether's Theorem. However there is a known special class of first integrals of nonholonomic systems generated by vector fields tangent to the group orbits, called {\it horizontal gauge momenta}, that suggest that some version of this link should still hold. In this paper we prove that, under certain conditions on the symmetry Lie group, the (nonholonomic) momentum map is conserved along the nonholonomic dynamics, thus extending Noether 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. In particular, for the snakeboard we show the existence of a new horizontal gauge momentum that reveals new aspects of its integrability.