  In this paper, we study the stability property of Hamiltonian systems on the
Wasserstein space. Let $H$ be a given Hamiltonian satisfying certain
properties. We regularize $H$ using the Moreau-Yosida approximation and denote
it by $H_\tau.$ We show that solutions of the Hamiltonian system for $H_\tau$
converge to a solution of the Hamiltonian system for $H$ as $\tau$ converges to
zero.
  We provide sufficient conditions on $H$ to carry out this process.
