  In some previous papers, a geometric description of Lagrangian Mechanics on
Lie algebroids has been developed. In the present paper, we give a Hamiltonian
description of Mechanics on Lie algebroids. In addition, we introduce the
notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove
that the Lagrangian (Hamiltonian) dynamics on Lie algebroids may be described
in terms of Lagrangian submanifolds of symplectic Lie algebroids. The
Lagrangian (Hamiltonian) formalism on Lie algebroids permits to deal with
Lagrangian (Hamiltonian) functions not defined necessarily on tangent
(cotangent) bundles. Thus, we may apply our results to the projection of
Lagrangian (Hamiltonian) functions which are invariant under the action of a
symmetry Lie group. As a consequence, we obtain that Lagrange-Poincare
(Hamilton-Poincare) equations are the Euler-Lagrange (Hamilton) equations
associated with the corresponding Atiyah algebroid. Moreover, we prove that
Lagrange-Poincare (Hamilton-Poincare) equations are the local equations
defining certain Lagrangian submanifolds of symplectic Atiyah algebroids.
