The Navier-Stokes equation or the Euler equation are usually derived as the conservation laws.

However, I wonder if there exists a Lagrangian $L$ or equivalently, an action functional $S[\overrightarrow{v}]$ of velocity vector fields $\overrightarrow{v}$ which yield the NS / Euler equations as the equation of motion. By the equation of motion, I mean the Euler-Lagrange equation.

Also, is it possible to realize the incompressibility condition $\nabla \cdot \overrightarrow{v}=0$ as a constraint by means of some Lagrange multiplier as well?

Could anyone please provide relevant reference, or the form of such action $S$?

This post imported from StackExchange Physics at 2024-06-18 14:50 (UTC), posted by SE-user Keith