First integrals of non-holonomic systems and their generators

Abstract

We discuss various aspects of mechanical systems with general (nonlinear) non-holonomic constraints from the perspective of presymplectic geometry. We begin by introducing a 2-form on the evolution space of a system having the property, among others, of modelling the unconstrained dynamics. Using this 2-form we then characterize a unique second-order dynamics on the constraint submanifold through a simple geometrical implementation of Chetaev's concept of virtual work. We also give necessary and sufficient conditions in order for the reduced dynamics to admit a non-holonomic Lagrangian formulation. Finally, we study the structure of a set of vector fields on the constraint submanifold which generates all first integrals of a constrained system. The relationships with a previously proposed set of vector fields in non-conservative holonomic mechanics and with known generalizations of Noether's theorem for non-holonomic systems are analysed.