Hamiltonian fluid mechanics

Hamiltonial fluid mechanics is the application of Hamiltonian methods to fluid mechanics. This formalism can only apply to nondissipative fluids for obvious reasons.

Take the simple example of a barotropic, inviscid vorticity-free fluid.

Then, the conjugate fields are the density field ρ and the velocity potential φ. The Poisson bracket is given by

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

and the Hamiltonian by

$H=\int d^dx \left[ \frac{1}{2}\rho(\nabla \phi)^2 +u(\rho) \right]$

where u is the internal energy density.

This gives rise to the following two equations of motion:

$\frac{\partial \rho}{\partial t}=-\nabla\cdot(\rho\vec{v})$

$\frac{\partial \phi}{\partial t}=\frac{1}{2}v^2+u'$

where Failed to parse (unknown function\stackrel): \vec{v}\ \stackrel{\mathrm{def}}{=}\ -\nabla \phi

is the velocity and is vorticity-free. The second equation leads to the Euler equations

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

after exploiting the fact that the vorticity is zero.

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Hamiltonian fluid mechanics

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