Hamiltonian fluid mechanics
Hamiltonian fluid mechanics is the application of Hamiltonian methods to
fluid mechanics. This formalism can only apply to non dissipativefluids.
Irrotational barotropic flow
Take the simple example of a
barotropic, inviscid vorticity-freefluid.
Then, the conjugate fields are the
mass densityfield "ρ" and the velocity potential"φ". The Poisson bracketis given by
where "e" is the
internal energydensity, as a function of "ρ". For this barotropic flow, the internal energy is related to the pressure "p" by:
where an apostrophe ('), denotes differentiation with respect to "ρ".
This Hamiltonian structure gives rise to the following two
equations of motion:
where is the velocity and is
vorticity-free. The second equation leads to the Euler equations:
after exploiting the fact that the
Luke's variational principle
*cite journal | journal=Annual Review of Fluid Mechanics | volume=20 | pages=225–256 | year=1988 | doi=10.1146/annurev.fl.20.010188.001301 | title=Hamiltonian Fluid Mechanics | author=R. Salmon
*cite journal | title=Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics | author=T. G. Shepherd | year=1990 | journal=Advances in Geophysics | volume=32 | pages=287–338
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