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.

43 pages International journal of Geometric Methods in Modern Physics, 2013. arXiv admin note: substantial text overlap with arXiv:1108.2844, arXiv:1007.1541, arXiv:1109.1242, arXiv:1101.0960, arXiv:1108.5050