Momentum map reduction for nonholonomic systems

Abstract

This paper presents a reduction procedure for nonholonomic systems admitting suitable types of symmetries and conserved quantities. The full procedure contains two steps. The first (simple) step results in a Chaplygin system, described by an almost symplectic structure, carrying additional symmetries. The focus of this paper is on the second step, which consists of a Marsden–Weinstein–type reduction that generalises constructions in (Balseiro and Fernandez 2015 Nonlinearity 28 2873–912, Cortés Monforte 2002 Geometric, Control and Numerical Aspects of non-Holonomic Systems (Springer)). The almost symplectic manifolds obtained in the second step are proven to coincide with the leaves of the reduced nonholonomic brackets defined in (Balseiro and Yapu-Quispe 2021 Ann. Inst. Henri Poincare C 38 23–60). We illustrate our construction with several classical examples.