  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.
