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Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

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Abstract

This paper studies the construction of geometric integrators for nonholonomic systems. We develop a formalism for nonholonomic discrete Euler–Lagrange equations in a setting that 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).

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Authors and Affiliations

  1. Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006, Madrid, Spain

    David Iglesias & David Martín de Diego

  2. Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain

    Juan C. Marrero

  3. Departamento de Matemática Aplicada and IUMA, Facultad de Ciencias, Universidad de Zaragoza, 50009, Zaragoza, Spain

    Eduardo Martínez

Authors
  1. David Iglesias
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  2. Juan C. Marrero
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  3. David Martín de Diego
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  4. Eduardo Martínez
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Corresponding author

Correspondence to David Iglesias.

Additional information

This work was partially supported by MEC (Spain) Grants MTM 2006-03322, MTM 2007-62478, MTM 2006-10531, project “Ingenio Mathematica” (i-MATH) No. CSD 2006-00032 (Consolider-Ingenio 2010) and S-0505/ESP/0158 of the CAM.

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Iglesias, D., Marrero, J.C., de Diego, D.M. et al. Discrete Nonholonomic Lagrangian Systems on Lie Groupoids. J Nonlinear Sci 18, 221–276 (2008). https://doi.org/10.1007/s00332-007-9012-8

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