Inverse problem for Lagrangian systems on Lie algebroids and applications to reduction by symmetries

TL;DR

This paper extends the inverse calculus of variations to Lie algebroids using Lagrangian submanifolds, providing new Helmholtz conditions and applications to symmetry reduction, with examples and comparisons to existing methods.

Contribution

It introduces a geometric framework for the inverse problem on Lie algebroids and derives Helmholtz conditions for Atiyah algebroids, linking inverse problems with symmetry reduction.

Findings

Helmholtz conditions on Lie algebroids are necessary but not sufficient.

New Helmholtz conditions for Atiyah algebroids are established.

Examples demonstrate the applicability and comparison with previous approaches.

Abstract

The language of Lagrangian submanifolds is used to extend a geometric characterization of the inverse problem of the calculus of variations on tangent bundles to regular Lie algebroids. Since not all closed sections are locally exact on Lie algebroids, the Helmholtz conditions on Lie algebroids are necessary but not sufficient, so they give a weaker definition of the inverse problem. As an application the Helmholtz conditions on Atiyah algebroids are obtained so that the relationship between the inverse problem and the reduced inverse problem by symmetries can be described. Some examples and comparison with previous approaches in the literature are provided.