Hamiltonian fluid mechanics

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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 \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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