Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

Abstract

This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie groupoid. We also discuss the existence of nonholonomic evolution operators in terms of the discrete nonholonomic Legendre transformations and in terms of adequate decompositions of the prolongation of the Lie groupoid. The characterization of the reversibility of the evolution operator and the discrete nonholonomic momentum equation are also considered. Finally, we illustrate with several classical examples the wide range of application of the theory (the discrete nonholonomic constrained particle, the Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on a rotating table and the two wheeled planar mobile robot).

45 pages

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• Higher Order Variational Integrators: a polynomial approach
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• Feedback Integrators for Nonholonomic Mechanical Systems
• Nonassociative analogs of Lie groupoids
• Matched pairs of discrete dynamical systems
• Discrete mechanics on unitary octonions
• The geometric discretisation of the Suslov problem: a case study of consistency for nonholonomic integrators
• The leaf space of a multiplicative foliation
• Lagrangian Mechanics and Reduction on Fibered Manifolds
• Lagrangian reduction of nonholonomic discrete mechanical systems by stages
• Singular lagrangian systems and variational constrained mechanics on Lie algebroids
• Lagrangian reduction of nonholonomic discrete mechanical systems