Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods

Pages: 220 - 229, Issue Special, September 2009

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Sebastián J. Ferraro - Departamento de Mateemática and Instituto de Matemática Bahía Blanca, Universidad Nacional del Sur, Av. Alem 1253, 8000 Bahía Blanca and CONICET, Argentina (email)
David Iglesias-Ponte - Unidad asociada ULL-CSIC “Geometría Diferencial y Mecánica Geométrica”, Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain (email)
D. Martín de Diego - Unidad Asociada ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Serrano 123, 28006 Madrid, Spain (email)

Abstract: Here we discuss a geometric integrator for nonholonomic mechanical systems that preserves the nonholonomic constraints, the discrete nonholonomic momentum map, and is also energy-preserving in some important cases. This method does not require a predefined discretization of the nonholonomic constraints. In Euclidean space, it yields a generalization of the classical SHAKE and RATTLE algorithms to the nonholonomic setting. This article shows that the method is second order convergent.

Keywords:  RATTLE method, holonomic and nonholonomic systems, variational integrators, energy-momentum integrators
Mathematics Subject Classification:  37J60, 37M15, 70F25

Received: August 2008;      Revised: April 2009;      Published: September 2009.