Dirac structures and Hamilton-Jacobi theory for Lagrangianmechanics on Lie algebroids

Melvin Leok; Diana Sosa

doi:10.3934/jgm.2012.4.421

Article Contents

2012, Volume 4, Issue 4: 421-442. Doi: 10.3934/jgm.2012.4.421

This issue Previous Article Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds Next Article The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity

Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids

Melvin Leok^1, and
Diana Sosa^2,

1.
Department of Mathematics, University of California, San Diego, 9500 Gilman Drive La Jolla, CA 92093-0112
2.
Departamento de Economía Aplicada y Unidad Asociada ULL-CSIC, "Geometría Diferencial y Mecánica Geométrica," Facultad de CC. EE. y Empresariales, Universidad de La Laguna, and Universidad Europea de Canarias, Calle de Inocencio García 1, La Orotava, Tenerife, Canary Islands

Received: May 31, 2012

Revised: October 31, 2012

Published: November 2012

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Abstract

This paper develops the notion of implicit Lagrangian systems on Lie algebroids and a Hamilton--Jacobi theory for this type of system. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of an implicit Lagrangian system on a Lie algebroid $E$ using Dirac structures on the Lie algebroid prolongation $\mathcal{T}^E E^*$. This setting includes degenerate Lagrangian systems with nonholonomic constraints on Lie algebroids.

Keywords:

Mathematics Subject Classification: Primary: 37J60, 53D17, 70H20; Secondary: 17B66, 70F25, 70G45, 70H45.

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