Lagrangian Lie subalgebroids generating dynamics for second-order mechanical systems on Lie algebroids

TL;DR

This paper develops a geometric framework for deriving the dynamics of second-order mechanical systems on Lie algebroids using Lagrangian Lie subalgebroids and Tulczyjew's triple, unifying various mechanical systems.

Contribution

It introduces a method to derive dynamics via Lagrangian Lie subalgebroids on symplectic Lie algebroids, extending the geometric formalism for mechanics on Lie algebroids.

Findings

Constructs a Lagrangian Lie subalgebroid for second-order systems

Utilizes Tulczyjew's triple in the Lie algebroid setting

Provides a unified geometric approach for diverse mechanical systems

Abstract

The study of mechanical systems on Lie algebroids permits an understanding of the dynamics described by a Lagrangian or Hamiltonian function for a wide range of mechanical systems in a unified framework. Systems defined in tangent bundles, Lie algebras, principal bundles, reduced systems and constrained are included in such description. In this paper, we investigate how to derive the dynamics associated with a Lagrangian system defined on the set of admissible elements of a given Lie algebroid using Tulczyjew's triple on Lie algebroids and constructing a Lagrangian Lie subalgebroid of a symplectic Lie algebroid, by building on the geometric formalism for mechanics on Lie algebroids developed by M. de Le\'on, J.C. Marrero and E. Mart\'inez on "Lagrangian submanifolds and dynamics on Lie algebroids".