Hydrostatic equilibrium and Navier-Stokes equations

[tom.stoer]

Is it possible to derive the condition for hydrostatic equilibrium or the Navier-Stokes equation for a self-gravitating fluid - e.g. for water on a planet with non-homogeneous density - based on a variational principle?

(the planet itself is assumed to be a fixed hard core not subject to the variation)

 

[Haborix]

My answer is I don't know. But I do know there is work on the Hamiltonian structure of fluids (see link). I also imagine relativists have work on fluids coupled to gravity in a Lagrangian formulation. However, since you want to produce Navier-Stokes which includes viscosity I worry that might prevent the use of a variational principle.

 

[tom.stoer]

My answer is I don't know.

Hm, are there other replies than just that?

But I know there is work on the Hamiltonian structure of fluids (see link).

Thanks a lot. I'll check that as soon as possible.

However, since you want to produce Navier-Stokes which includes viscosity I worry that might prevent the use of a variational principle.

Not necessarily including viscosity. I would be glad to see any variational principle. So let's discuss with the Euler equations instead.