Nonholonomic momentum map reduction and a Chaplygin-type foliation

Abstract This paper presents a set-up for momentum map reduction of nonholonomic systems with symmetries, extending previous constructions in Balseiro et al (2023 Nonlinearity 36 5401); Cortés Monforte (2002 Geometric, Control and Numeric Aspects of Nonholonomic Systems (Springer)), based on the existence of certain conserved quantities and making essential use of the nonholonomic momentum bundle map of Bloch et al (1996 Arch. Ration. Mech. Anal. 136 21–99). We show that the reductions of the momentum level sets carry an almost symplectic form codifying the reduced dynamics. These reduced manifolds are the leaves of the foliation associated with an almost Poisson bracket obtained by a (dynamically compatible) gauge transformation of the nonholonomic bracket. We show that each leaf is a Chaplygin-type leaf , in the sense that it is isomorphic to a cotangent bundle with the canonical symplectic form plus a ‘magnetic term’.