8
$\begingroup$

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 $S$?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Related: physics.stackexchange.com/q/14652/2451 $\endgroup$
    – Qmechanic ♦
    Commented May 19, 2022 at 16:53
  • $\begingroup$ One reference to look at is arxiv.org/abs/0810.0817 See also Chapter 13 of "Variational Principles In Dynamics And Quantum Theory" and references therein. Also see the authors Yourgrau and Mandelstam. It appears that dissipative systems of equations don't seem to be well represented by a variation of an action. $\endgroup$
    – John Bussoletti
    Commented Dec 29, 2023 at 1:37

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.