  We use a new variational method --based on the theory of anti-selfdual
Lagrangians developed in [2] and [3]-- to establish the existence of solutions
of convex Hamiltonian systems that connect two given Lagrangian submanifolds in
$\R^{2N}$. We also consider the case where the Hamiltonian is only semi-convex.
A variational principle is also used to establish existence for the
corresponding Cauchy problem. The case of periodic solutions will be considered
in a forthcoming paper [5].
