Lagrangian submanifolds and dynamics on Lie algebroids

In some previous papers, a geometric description of Lagrangian mechanics on Lie algebroids has been developed. In this topical review, 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 us 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–Poincaré (Hamilton–Poincaré) equations are the Euler–Lagrange (Hamilton) equations associated with the corresponding Atiyah algebroid. Moreover, we prove that Lagrange–Poincaré (Hamilton–Poincaré) equations are the local equations defining certain Lagrangian submanifolds of symplectic Atiyah algebroids.