Momentum and Energy Preserving Integrators for Nonholonomic Dynamics

In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian Ld : Q × Q → R, where Q is the configuration manifold, and a (generally nonintegrable) distribution D ⊂ TQ. In the proposed method, a discretization of the constraints is not required. We show that the method preserves the discrete nonholonomic momentum map, and also that the nonholonomic constraints are preserved in average. We study in particular the case where Q has a Lie group structure and the discrete Lagrangian and/or nonholonomic constraints have various invariance properties, and show that the method is also energy-preserving in some important cases.