  We study discretizations of Hamiltonian systems on the probability density
manifold equipped with the $L^2$-Wasserstein metric. Based on discrete optimal
transport theory, several Hamiltonian systems on graph (lattice) with different
weights are derived, which can be viewed as spatial discretizations to the
original Hamiltonian systems. We prove the consistency and provide the
approximate orders for those discretizations. By regularizing the system using
Fisher information, we deduce an explicit lower bound for the density function,
which guarantees that symplectic schemes can be used to discretize in time.
Moreover, we show desirable long time behavior of these schemes, and
demonstrate their performance on several numerical examples.
