Hamiltonian Dynamics on Lie Algebroids, Unimodularity and Preservation of Volumes

TL;DR

This paper explores how the unimodularity of Lie algebroids influences the existence of invariant volume forms in Hamiltonian dynamics, with applications to various examples of Lie algebroids.

Contribution

It establishes a connection between unimodularity of Lie algebroids and invariant volume forms in Hamiltonian systems, providing new insights and applications.

Findings

Unimodularity of Lie algebroids is linked to the existence of invariant volume forms.

Results are applied to multiple Hamiltonian systems on different Lie algebroids.

The paper advances understanding of volume preservation in Lie algebroid dynamics.

Abstract

In this paper we discuss the relation between the unimodularity of a Lie algebroid $\tau_{A}: A\to Q$ and the existence of invariant volume forms for the hamiltonian dynamics on the dual bundle $A^{*}$. The results obtained in this direction are applied to several hamiltonian systems on different examples of Lie algebroids.