Hamiltonian fluid mechanics

Hamiltonian fluid mechanics is the application of Hamiltonian methods to fluid mechanics. Note that this formalism only applies to non-dissipative 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

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

and the Hamiltonian by:

$H=\int {\mathrm{d}}^{d}x{\mathscr{H}}=\int {\mathrm{d}}^{d}x(\frac{1}{2}\rho (\mathrm{\nabla }\phi {)}^2+e(\rho )),$

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:

$e^{″}=\frac{1}{\rho }{p}^{\prime },$

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

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

$\begin{array}{rl}\frac{\partial \rho }{\partial t}& =+\frac{\partial {\mathscr{H}}}{\partial \phi }=-\mathrm{\nabla }\cdot (\rho \overrightarrow{u}),\\ \frac{\partial \phi }{\partial t}& =-\frac{\partial {\mathscr{H}}}{\partial \rho }=-\frac{1}{2}\overrightarrow{u}\cdot \overrightarrow{u}-e^{\prime },\end{array}$

where $\overrightarrow{u}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\mathrm{\nabla }\phi$ is the velocity and is vorticity-free. The second equation leads to the Euler equations:

$\frac{\partial \overrightarrow{u}}{\partial t}+(\overrightarrow{u}\cdot \mathrm{\nabla })\overrightarrow{u}=-e^{″}\mathrm{\nabla }\rho =-\frac{1}{\rho }\mathrm{\nabla }p$

after exploiting the fact that the vorticity is zero:

$\mathrm{\nabla }\times \overrightarrow{u}=\overrightarrow{0}.$

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

See also

• Luke's variational principle
• Hamiltonian field theory