Lagrangian Lie subalgebroids of the canonical symplectic Lie algebroid

It is well-known that the Poisson reduction of a hamiltonian system on the cotangent bundle of a manifold produces a hamiltonian system on a linear Poisson manifold. On the other hand, linear Poisson structures on a vector bundle $A^*$ may be described in terms of the canonical symplectic section of ${\mathcal T}^AA^*$, the $A$-tangent bundle to $A^*$. In fact, ${\mathcal T}^AA^*$ is a canonical symplectic Lie algebroid over the linear Poisson manifold $A^*$. In this master thesis, we discuss Lagrangian Lie subalgebroids of ${\mathcal T}^AA^*$. The base space of a Lagrangian Lie subalgebroid $L$ turns out to be a coisotropic submanifold $C$ of $A^*$. Thus, first we describe the local nature of $C$ when it is an affine subbundle of $A^*$ and then we describe the local nature of $L$. We expect that these results may be applied, in a future work, in the geometric formulation of Hamilton-Jacobi theory for reduced hamiltonian systems