Hamiltonian Mechanics on Duals of Generalized Lie Algebroids

TL;DR

This paper introduces a novel Hamiltonian mechanics framework on duals of generalized Lie algebroids, developing geometric structures and equations, including Hamilton-Jacobi equations, extending classical mechanics.

Contribution

It presents a new geometric formalism for Hamiltonian mechanics on duals of generalized Lie algebroids, including dual mechanical systems and associated structures.

Findings

Defined dual mechanical, Hamiltonian, and Cartan systems on Lie algebroids.

Derived canonical semisprays and Hamilton-Jacobi equations for these systems.

Extended classical Hamiltonian formalism to a generalized Lie algebroid setting.

Abstract

A new description, different by the classical theory of Hamiltonian Mechanics, in the general framework of generalized Lie algebroids is presented. In the particular case of Lie algebroids, new and important results are obtained. We present the \emph{dual mechanical systems} called by use, \emph{dual mechanical}$(\rho,\eta) $\emph{-systems, Hamilton mechanical}$(\rho,\eta) $\emph{-systems} or \emph{% Cartan mechanical}$(\rho,\eta) $\emph{-systems.} and we develop their geometries. We obtain the canonical $(\rho,\eta) $\emph{-}semi(spray) associated to a dual mechanical $(\rho,\eta) $-system. The Hamilton mechanical $(\rho,\eta)$-systems are the spaces necessary to develop a Hamiltonian formalism. We obtain the $(\rho,\eta)$-semispray associated to a regular Hamiltonian $H$ and external force $F_{e}$ and we derive the equations of Hamilton-Jacobi type.