In 1982, following P.J. Morrison, J.E. Marsden and A. Weinstein introduced a Poisson structure on an ”infinite dimensional manifold”. This allows them to interpret systems such as the Maxwell–Poisson, as infinite dimensional Hamiltonian systems. In this talk, we consider physical systems where electric and magnetic fields are absent. In that case, their manifold reduces to M, the set of Borel probability measures on IR2d of bounded second moment. One obtains a foliation of
M and a differentiable structure defined only on each leaf of the foliation. Solutions of Hamiltonian ODEs are paths moving within the closure of each leaf. We establish rigorous results such as existence of solutions for Hamiltonian ODEs. We prove Green’s formula for annular ”surfaces” of M, without requiring smoothness properties. (Talk based on joint works with L. Ambrosio and Kim–Pacini).