Mechanical Systems in the Generalized Lie Algebroids Framework

Abstract

\emph{Mechanical systems} called by use, \emph{mechanical}$\left(\rho ,\eta\right) $\emph{-systems, Lagrange mechanical}$\left(\rho ,\eta \right) $\emph{-systems} or \emph{Finsler mechanical}$\left(\rho ,\eta \right) $\emph{-systems} are presented. The canonical $\left(\rho ,\eta \right) $\emph{-}semi(spray) associated to a mechanical $\left(\rho ,\eta \right) $-system is obtained. New and important results are obtained in the particular case of Lie algebroids. The Lagrange mechanical $(\rho ,\eta)$% -systems are the spaces necessary to develop a new Lagrangian formalism. We obtain the $(\rho ,\eta)$-semispray associated to a regular Lagrangian $L$ and external force $F_{e}$ and we derive the equations of Euler-Lagrange type. So, a new solution for the Weinstein's Problem in the general framework of generalized Lie algebroids is presented.