# Fluid Mechanics from a variational principle

It is possible to define a good variational principle to describe Fluid Mechanics? if so, what is the correct treatment of the issue. I guess something like:

$$I=\int d^4x \left(\frac{1}{2}\rho v^2-P-\rho g x\right)$$

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You should probably be more specific about what equation you want to reproduce. The full Navier-Stokes equations? Incompressible? Euler equations? Any of the above? Also, I know its just a first attempt, but the equation you wrote down has many flaws. – BebopButUnsteady Sep 14 '11 at 16:00
I'm not an expert in the field, but I found a reference that claims this is possible. prl.aps.org/abstract/PRL/v58/i14/p1419_1 (behind a PRL paywall). – Gerben Sep 14 '11 at 17:29
Thanks very much you both. BebopButUnsteady You are rigth, i meant Euler's equation. – Adolfo_Toloza Sep 15 '11 at 14:06
A nice lagrangian treatment is available for incompressible potential flows with hydrostatic pressure, or for flows allowing one component of vorticity. Also, for a general Hamiltonian description one may resort to Clebsch variables, and then define the resultant lagrangian from these. One can check out some of Zakharov's reviews on the subject. – user8260 Apr 14 '12 at 22:27