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 the velocity potential φ. The Poisson bracket is given by

$\{\varphi ({\vec {x}}),\rho ({\vec {y}})\}=\delta ^{d}({\vec {x}}-{\vec {y}})$

and the Hamiltonian by:

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

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:

$e''={\frac {1}{\rho }}p',$

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

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

${\begin{aligned}{\frac {\partial \rho }{\partial t}}&=+{\frac {\delta {\mathcal {H}}}{\delta \varphi }}=-{\vec {\nabla }}\cdot (\rho {\vec {v}}),\\{\frac {\partial \varphi }{\partial t}}&=-{\frac {\delta {\mathcal {H}}}{\delta \rho }}=-{\frac {1}{2}}{\vec {v}}\cdot {\vec {v}}-e',\end{aligned}}$

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

${\frac {\partial {\vec {v}}}{\partial t}}+({\vec {v}}\cdot \nabla ){\vec {v}}=-e''\nabla \rho =-{\frac {1}{\rho }}\nabla {p}$

after exploiting the fact that the vorticity is zero:

${\vec {\nabla }}\times {\vec {v}}={\vec {0}}.$

See also

• Luke's variational principle

References

• R. Salmon (1988). "Hamiltonian Fluid Mechanics". Annual Review of Fluid Mechanics. 20 (1): 225–256. doi:10.1146/annurev.fl.20.010188.001301.
• T. G. Shepherd (1990). "Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics". Advances in Geophysics. 32: 287–338. doi:10.1016/S0065-2687(08)60429-X.