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

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 TEE∗. This setting includes degenerate Lagrangian systems with nonholonomic constraints on Lie algebroids.