Conserved quantities and Hamiltonization of nonholonomic systems

Abstract

This paper studies hamiltonization of nonholonomic systems using geometric tools, building on [1,5]. The main novelty in this paper is the use of symmetries and suitable first integrals of the system to explicitly define a new bracket on the reduced space that codifies the nonholonomic dynamics and carries, additionally, an almost symplectic foliation (determined by the common level sets of the first integrals); in particular cases of interest, this new bracket is a Poisson structure that hamiltonizes the system. Our construction of the new bracket is based on a gauge transformation of the nonholonomic bracket by a global 2-form that we explicitly describe. We study various geometric features of the reduced brackets and apply our formulas to obtain a geometric proof of the hamiltonization of a homogeneous ball rolling without sliding in the interior side of a convex surface of revolution.