  Anti-selfdual Lagrangians on a state space lift to path space provided one
adds a suitable selfdual boundary Lagrangian. This process can be iterated by
considering the path space as a new state space for the newly obtained
anti-selfdual Lagrangian. We give here two applications for these remarkable
permanence properties. In the first, we establish for certain convex-concave
Hamiltonians ${\cal H}$ on a --possibly infinite dimensional--symplectic space
$H^2$, the existence of a solution for the Hamiltonian system $-J\dot u
(t)=\partial {\cal H} (u(t))$ that connects in a given time T>0, two Lagrangian
submanifolds. Another application deals with the construction of a
multiparameter gradient flow for a convex potential. Our methods are based on
the new variational calculus for anti-selfdual Lagrangians developed in [4],
[5] and [7].
