Application of Stochastic Variational Method to Hydrodynamics

TL;DR

This paper demonstrates how the stochastic variational method can derive the Navier-Stokes equation for ideal fluids, linking dissipation effects to fluctuation dissipation theorem, with a more general Lagrangian formulation and Hamiltonian approach.

Contribution

It introduces a more general parameterization of the Lagrangian in the stochastic variational method and shows consistent results using Hamiltonian variation.

Findings

Derivation of Navier-Stokes equation via SVM

Reformulation of transport coefficients

Equivalence of Lagrangian and Hamiltonian approaches

Abstract

We apply the stochastic variational method to the action of the ideal fluid and showed that the Navier-Stokes equation is derived. In this variational method, the effect of dissipation is realized as the direct consequence of the fluctuation dissipation theorem. Differently from the previous works \cite{kk1,kk2}, we parameterize the Lagrangian of SVM in more general form. The form of the obtained equation is not modified but the definition of the transport coefficients are changed. We further discuss the formulation of SVM using the Hamiltonian and show that the variation of the Hamiltonian gives the same result as the case of the Lagrangian.