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# 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 "&rho;" and the velocity potential "&phi;". The Poisson bracket is given by

:$\left\{varphi\left(vec\left\{x\right\}\right), ho\left(vec\left\{y\right\}\right)\right\}=delta^d\left(vec\left\{x\right\}-vec\left\{y\right\}\right)$

and the Hamiltonian by:

:$mathcal\left\{H\right\}=int mathrm\left\{d\right\}^d x left \left[ frac\left\{1\right\}\left\{2\right\} ho\left(vec\left\{ abla\right\} varphi\right)^2 +e\left( ho\right) ight\right] ,$

where "e" is the internal energy density, as a function of "&rho;". For this barotropic flow, the internal energy is related to the pressure "p" by:

:$e" = frac\left\{1\right\}\left\{ ho\right\}p\text{'},$

where an apostrophe ('), denotes differentiation with respect to "&rho;".

This Hamiltonian structure gives rise to the following two equations of motion:

:

where $vec\left\{v\right\} stackrel\left\{mathrm\left\{def\left\{=\right\} abla varphi$ is the velocity and is vorticity-free. The second equation leads to the Euler equations:

:$frac\left\{partial vec\left\{v\left\{partial t\right\} + \left(vec\left\{v\right\}cdot abla\right) vec\left\{v\right\} = -e" abla ho = -frac\left\{1\right\}\left\{ ho\right\} abla\left\{p\right\}$

after exploiting the fact that the vorticity is zero:

:$vec\left\{ abla\right\} imesvec\left\{v\right\}=vec\left\{0\right\}.$

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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