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

Melvin Leok,Diana Sosa

doi:10.3934/jgm.2012.4.421

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.

Highlights

There is a vast literature on the Lagrangian formalism in mechanics, which is due to the central role played by these systems in the foundations of modern mathematics and physics
We introduce the notion of an implicit Lagrangian system on a Lie algebroid E using the induced generalized Dirac structure DU on the Lie algebroid prolongation TEE∗ → (TEE)∗ that is naturally induced by a vector subbundle U of E and we obtain the Hamilton–Jacobi theorem for this kind of systems
We introduced the notion of an induced almost Dirac structure, and show how it leads to implicit Lagrangian systems on Lie algebroids

Summary

Introduction

There is a vast literature on the Lagrangian formalism in mechanics, which is due to the central role played by these systems in the foundations of modern mathematics and physics. Problems often arise due to their singular nature, which gives rise to constraints that address the fact that the evolution problem is not well-posed (internal constraints). Constraints can manifest a priori restrictions on the states of the system which arise due to physical arguments or from external conditions (external constraints). The more frequent case appears in the Lagrangian formalism of singular mechanical systems which are commonplace in many physical theories (as in Yang-Mills theories, gravitation, etc)

Objectives

Conclusion