Lagrangian Mechanics on Generalized Lie Algebroids

TL;DR

This paper develops a geometric framework for Lagrangian mechanics on generalized Lie algebroids, introducing new types of mechanical systems and deriving their equations of motion.

Contribution

It introduces generalized Lie algebroid-based mechanical systems and derives their Lagrangian formalism and equations of motion, extending classical mechanics.

Findings

Defined new mechanical systems on generalized Lie algebroids

Derived Euler-Lagrange equations in this framework

Connected geometric structures to classical mechanics

Abstract

A solution for the Weinstein's Problem in the general framework of generalized Lie algebroids is the target of this paper. We present the mechanical systems called by use, mechanical (?; ?)-systems, Lagrange mechanical (?; ?)-systems or Finsler mechanical (?; ?)-systems and we develop their geometries. We obtain the canonical (?; ?)-semi(spray) associated to a mechanical (?; ?)-system. The Lagrange mechanical (?; ?)-systems are the spaces necessary to develop a Lagrangian formalism. We obtain the (?; ?)-semispray associated to a regular Lagrangian L and external force Fe and we derive the equations of Euler-Lagrange type. A new point of view over classical and modern results about Lagrangian Mechanics are presented.