Nonholonomic Lagrangian systems on Lie algebroids

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June 2009, 24(2): 213-271. doi: 10.3934/dcds.2009.24.213

Nonholonomic Lagrangian systems on Lie algebroids

Jorge Cortés ^1, , Manuel de León ^2, , Juan Carlos Marrero ^3, and Eduardo Martínez ^4,

1.	Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 92093, United States
2.	Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, 28006 Madrid, Spain
3.	Departamento de Matemática Fundamental, Unidad Asociada ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Universidad de la Laguna, Tenerife, Canary Islands, Spain
4.	IUMA and Departamento de Matemática Aplicada, Universidad de Zaragoza, 50009 Zaragoza, Spain

Received May 2008 Revised September 2008 Published March 2009

Abstract
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This paper presents a geometric description on Lie algebroids of Lagrangian systems subject to nonholonomic constraints. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitable projection of the unconstrained dynamics. The proposed novel formalism provides new insights into the geometry of nonholonomic systems, and allows us to treat in a unified way a variety of situations, including systems with symmetry, morphisms, reduction, and nonlinearly constrained systems. Various examples illustrate the results.

Keywords: Lie algebroids, symmetry, Lagrange-d'Alembert equations, reduction., Nonholonomic Mechanics.

Mathematics Subject Classification: Primary: 70F25, 70H03, 70H33; Secondary: 37J60, 53D1.

Citation: Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213

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