Mechanical systems in the generalized Lie algebroids framework

Abstract

Mechanical systems, called by use, mechanical (ρ, η)-systems, Lagrange mechanical (ρ, η)-systems or Finsler mechanical (ρ, η)-systems, are presented. The canonical (ρ, η)-semi(spray) associated to a mechanical (ρ, η)-system is obtained. New and important results are obtained in the particular case of Lie algebroids. The Lagrange mechanical (ρ, η)-systems are the spaces necessary to develop a new Lagrangian formalism. We obtain the (ρ, η)-semispray associated to a regular Lagrangian L and external force Fe 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.