Contact formalism for dissipative mechanical systems on Lie algebroids

TL;DR

This paper develops a geometric framework for contact Lagrangian and Hamiltonian systems on Lie algebroids, extending contact geometry to dissipative systems and exploring Legendre transformations and Hamilton-Jacobi theory.

Contribution

It introduces a novel geometric description of dissipative systems on Lie algebroids using contact geometry, including Legendre transformations and Legendrian subalgebroids.

Findings

Established a contact geometric framework for dissipative systems on Lie algebroids

Connected Lagrangian and Hamiltonian formalisms via Legendre transformation

Explored Hamilton-Jacobi theory and Legendrian subalgebroids in this setting

Abstract

In this paper, we introduce a geometric description of contact Lagrangian and Hamiltonian systems on Lie algebroids in the framework of contact geometry, using the theory of prolongations. We discuss the relation between Lagrangian and Hamiltonian settings through a convenient notion of Legendre transformation. We also discuss the Hamilton-Jacobi problem in this framework and introduce the notion of a Legendrian Lie subalgebroid of a contact Lie algebroid.