Legendre Duality Between Lagrangian and Hamiltonian Mechanics

TL;DR

This paper explores the Legendre duality connecting Lagrangian and Hamiltonian mechanics within the framework of Lie algebroids, providing a unified geometric description and extending classical results.

Contribution

It develops a Legendre duality framework for Lagrangian and Hamiltonian mechanics using Lie algebroid structures, including new results for specific cases.

Findings

Establishment of a Legendre duality between Lagrangian and Hamiltonian mechanics.

Extension of classical mechanics results to Lie algebroids.

Unified geometric description of mechanical systems using ( ho, eta)-structures.

Abstract

In some previous papers, a Legendre duality between Lagrangian and Hamiltonian Mechanics has been developed. The (\rho,\eta)-tangent application of the Legendre bundle morphism associated to a Lagrangian L or Hamiltonian H is presented. Using that, a Legendre description of Lagrangian Mechanics and Hamiltonian Mechanics is developed. Duality between Lie algebroids structure, adapted (\rho,\eta)-basis, distinguished linear (\rho,\eta)-connections and mechanical (\rho,\eta)-systems is the scope of this paper. In the particular case of Lie algebroids, new results are presented. In the particular case of the usual Lie algebroid tangent bundle, the classical results are obtained.