  We discuss the interplay between lagrangian distributions and connections in
symplectic geometry, beginning with the traditional case of symplectic
manifolds and then passing to the more general context of poly- and
multisymplectic structures on fiber bundles, which is relevant for the
covariant hamiltonian formulation of classical field theory. In particular, we
generalize Weinstein's tubular neighborhood theorem for symplectic manifolds
carrying a (simple) lagrangian foliation to this situation. In all cases, the
Bott connection, or an appropriately extended version thereof, plays a central
role.
