# 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{align} \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{align}

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:

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