# Hamiltonian fluid mechanics

**Hamiltonian fluid mechanics**is the application of Hamiltonian methods tofluid 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:$\{varphi(vec\{x\}),\; ho(vec\{y\})\}=delta^d(vec\{x\}-vec\{y\})$

and the

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

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

where an apostrophe ('), denotes differentiation with respect to "ρ".

This Hamiltonian structure gives rise to the following two

equations of motion ::$egin\{align\}\; frac\{partial\; ho\}\{partial\; t\}=+frac\{deltamathcal\{H\{deltavarphi\}=\; -vec\{\; abla\}cdot(\; hovec\{v\}),\; \backslash \; frac\{partial\; varphi\}\{partial\; t\}=-frac\{deltamathcal\{H\{delta\; ho\}=-frac\{1\}\{2\}vec\{v\}cdotvec\{v\}-e\text{'},end\{align\}$

where $vec\{v\}\; stackrel\{mathrm\{def\{=\}\; abla\; varphi$ is the velocity and is

vorticity-free . The second equation leads to theEuler equations ::$frac\{partial\; vec\{v\{partial\; t\}\; +\; (vec\{v\}cdot\; abla)\; vec\{v\}\; =\; -e"\; abla\; ho\; =\; -frac\{1\}\{\; ho\}\; abla\{p\}$

after exploiting the fact that the

vorticity is zero::$vec\{\; abla\}\; imesvec\{v\}=vec\{0\}.$

**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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