On the discretization of nonholonomic dynamics in ℝn

Fernando Jiménez, Jürgen Scheurle

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3 Scopus citations

Abstract

In this paper we explore the nonholonomic Lagrangian setting of mechanical systems in coordinates on finite-dimensional configuration manifolds. We prove existence and uniqueness of solutions by reducing the basic equations of motion to a set of ordinary differential equations on the underlying distribution manifold D . Moreover, we show that any D-preserving discretization may be understood as being generated by the exact evolution map of a time-periodic non-autonomous perturbation of the original continuous-time nonholonomic system. By means of discretizing the corresponding Lagranged'Alembert principle, we construct geometric integrators for the original nonholonomic system. We give precise conditions under which these integrators generate a discrete flow preserving the distribution D . Also, we derive corresponding consistency estimates. Finally, we carefully treat the example of the nonholonomic particle, showing how to discretize the equations of motion in a reasonable way, particularly regarding the nonholonomic constraints.

Original languageEnglish
Pages (from-to)43-80
Number of pages38
JournalJournal of Geometric Mechanics
Volume7
Issue number1
DOIs
StatePublished - 1 Mar 2015

Keywords

  • Differential algebraic equations
  • Discrete variational calculus
  • Discretization as perturbation
  • Geometric integration
  • Nonholonomic mechanics
  • Ordinary differential equations

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