In this talk, we advocate an action function based variational approach to the hydrodynamic derivation of continuum mechanics equations. To illustrate, we use a Carleman type particle model and focus on stochastic hydrodynamics (instead of deterministic Hamiltonian particle models) where technical issues such as ergodicity is easier because of randomness in the model. However, we intentionally avoid the usual stochastic hydrodynamic approaches which heavily rely upon ergodic theories of Markov processes. We introduce a Hamiltonian formulation and introduce a weak KAM (Kolmogorov-Arnold-Moser) type argument to the derivation. A viscosity solution Hamilton-Jacobi theory is developed for the limiting effective Hamiltonian, which is defined in the space of probability measures. This is a joint work with Toshio Mikami and Johannes Zimmer.