- Hamiltonian fluid mechanics
Hamiltonian fluid mechanics is the application of Hamiltonian methods to
fluid mechanics . This formalism can only apply to nondissipative fluids.Irrotational barotropic flow
Take the simple example of a
barotropic ,inviscid vorticity-free fluid.Then, the conjugate fields are the
mass density field "ρ" and thevelocity potential "φ". ThePoisson bracket is given by:
and the
Hamiltonian by::
where "e" is the
internal energy density, 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 theEuler equations ::
after exploiting the fact that the
vorticity is zero::
ee also
*
Luke's variational principle References
*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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