Sobolev H1 geometry of the symplectomorphism group

For a closed symplectic manifold $(M,\omega )$ with compatible Riemannian metric g we study the Sobolev $H^1$ geometry of the group of all $H^s$ diffeomorphisms on M which preserve the symplectic structure. We show that, for sufficiently large s, the $H^1$ metric admits globally defined geodesics and the corresponding exponential map is a non-linear Fredholm map of index zero. Finally, we show that the $H^1$ metric carries conjugate points via some simple examples.