It is a theorem of Smale that the group of orientation-preserving diffeomorphisms of $D^2$, rel boundary, is contractible.  If the diffeomorphisms can move the boundary, you can establish a homotopy equivalence between that and the circle.  The diffeomorphisms do not have to preserve area.  Then, a [theorem of Moser][1] establishes a deformation retract from diffeomorphisms to volume-preserving diffeomorphisms.  Moser's result is easier to see if you have a closed manifold, but it extends to manifolds with boundary with the doubling trick.  Together, this indirectly gives you a curve of symplectomorphisms connecting the identity to $\psi$, since in two dimensions the symplectic structure is just a volume structure.  Finally if you have a smooth curve of area-preserving diffeomorphisms of a disk, I think there is a time-dependent Hamiltonian obtained by integrating the corresponding vector field.

  [1]: http://mathoverflow.net/questions/7817/normal-coordinates-for-a-manifold-with-volume-form