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.

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The geometry of the Lie algebroid generalized tangent bundle of a generalized Lie algebroid is developed. Formulas of Ricci type and identities of Cartan and Bianchi type are presented. Introducing the notion of geodesic of a mechanical (ρ, η)-system with respect to a (ρ, η)-spray, the Berwald (ρ, η)-derivative operator and its mixed curvature, we obtain main results to conceptualize the Weyl's method in this general framework. Finally, we obtain two new results of Weyl type for the geometry of mechanical (ρ, η)-systems.

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