I will report on progress of a joint project with Alan Weinstein in which we try to understand the initial value constraints of field theories with external symmetries as momenta of a Lie algebroid action on the presymplectic manifold of fields. Having shown earlier how a Lie algebroid symmetry naturally appears in General Relativity, we are currently studying a notion of hamiltonian momentum map for Lie algebroids. I will explain how hamiltonian Lie algebroids generally arise in lagrangian field theories with spacetime diffeomorphism symmetry.