Higher-order Variational Calculus on Lie algebroids

TL;DR

This paper develops a higher-order variational calculus framework on Lie algebroids, deriving critical point equations and exploring their relation to classical reduction methods like Euler-Poincaré equations.

Contribution

It introduces a novel higher-order variational calculus on Lie algebroids, connecting it with established reduction techniques and providing new insights into their geometric structure.

Findings

Derived equations for critical points of higher-order Lagrangians on Lie algebroids

Established relations with Euler-Poincaré and Lagrange Poincaré equations

Provided reduction and reconstruction results for these systems

Abstract

The equations for the critical points of the action functional defined by a Lagrangian depending on higher-order derivatives of admissible curves on a Lie algebroid are found. The relation with Euler-Poincar\'e and Lagrange Poincar\'e type equations is studied. Reduction and reconstruction results for such systems are established.