  Let M denote the space of probability measures on R^D endowed with the
Wasserstein metric. A differential calculus for a certain class of absolutely
continuous curves in M was introduced by Ambrosio, Gigli and Savare'. In this
paper we develop a calculus for the corresponding class of differential forms
on M. In particular we prove an analogue of Green's theorem for 1-forms and
show that the corresponding first cohomology group, in the sense of de Rham,
vanishes. For D=2d we then define a symplectic distribution on M in terms of
this calculus, thus obtaining a rigorous framework for the notion of
Hamiltonian systems as introduced by Ambrosio and Gangbo. Throughout the paper
we emphasize the geometric viewpoint and the role played by certain
diffeomorphism groups of R^D.
