Legendre Duality Between Lagrangian and Hamiltonian Mechanics

In some previous papers, a Legendre duality between Lagrangian and Hamiltonian Mechanics has been developed. The (\rho,\eta)-tangent application of the Legendre bundle morphism associated to a Lagrangian L or Hamiltonian H is presented. Using that, a Legendre description of Lagrangian Mechanics and Hamiltonian Mechanics is developed. Duality between Lie algebroids structure, adapted (\rho,\eta)-basis, distinguished linear (\rho,\eta)-connections and mechanical (\rho,\eta)-systems is the scope of this paper. In the particular case of Lie algebroids, new results are presented. In the particular case of the usual Lie algebroid tangent bundle, the classical results are obtained.