Wasserstein Hamiltonian flows

We establish kinetic Hamiltonian flows in density space embedded with the $L^2$-Wasserstein metric tensor. We derive the Euler-Lagrange equation in density space, which introduces the associated Hamiltonian flows. We demonstrate that many classical equations, such as Vlasov equation, Schr{"o}dinger equation and Schr{"o}dinger bridge problem, can be rewritten as the formalism of Hamiltonian flows in density space.