Exterior differential systems on Lie algebroids and the invariant inverse problem of the calculus of variations

TL;DR

This paper extends exterior differential systems to Lie algebroids, providing a framework for analyzing invariant inverse problems in the calculus of variations with applications to symmetric dynamical systems.

Contribution

It introduces the concept of integral manifolds on Lie algebroids and applies this theory to symmetry-invariant inverse problems in calculus of variations.

Findings

Extended exterior differential systems to Lie algebroids.

Defined integral manifolds in this new setting.

Applied theory to symmetric dynamical systems and inverse problems.

Abstract

We extend the theory of exterior differential systems from manifolds and their tangent bundles to Lie algebroids. In particular, we define the concept of an integral manifold of such an exterior differential system. We support our developments with several examples, including an application to dynamical systems with a symmetry group and to the invariant inverse problem of the calculus of variations.