Hamiltonian fluid mechanics

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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(\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{align}
  \frac{\partial \rho}{\partial t}&=+\frac{\partial \mathcal{H}}{\partial \varphi}= -\nabla \cdot(\rho\vec{u}),
  \\
  \frac{\partial \varphi}{\partial t}&=-\frac{\partial \mathcal{H}}{\partial \rho}=-\frac{1}{2}\vec{u}\cdot\vec{u}-e',
\end{align}

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

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

after exploiting the fact that the vorticity is zero:

\nabla \times\vec{u}=\vec{0}.

As fluid dynamics is described by non-canonical dynamics, which possess an infinite amount of Casimir invariants, an alternative formulation of Hamiltonian formulation of fluid dynamics can be introduced through the use of Nambu mechanics[1][2]

See also

Notes

References

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