Hamiltonian fluid mechanics

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

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

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

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.