  We study orbits and reachable sets of generic couples of Hamiltonians $H_1,
H_2$ on a symplectic manifold $N$. We prove that, $C^k$-generically for $k$
large enough, orbits coincide with the whole of $N$, and that the same is true
for reachable sets when $N$ is compact.
  Our results are stated in terms of a strong form of genericity which makes
use of the notion of rectifiable subsets of positive codimension in Banach or
Frechet spaces.
