Discrete Nonholonomic Lagrangian Systems on Lie Groupoids |
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Authors: | David Iglesias Juan C Marrero David Martín de Diego Eduardo Martínez |
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Institution: | (1) Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain;(2) Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain;(3) Departamento de Matemática Aplicada and IUMA, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain |
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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).
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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Keywords: | Discrete Mechanics Nonholonomic Mechanics Lie groupoids Lie algebroids Reduction Nonholonomic momentum map |
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