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

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

and the Hamiltonian by:

${\displaystyle H=\int \mathrm {d} ^{d}x{\mathcal {H))=\int \mathrm {d} ^{d}x\left({\frac {1}{2))\rho (\nabla \varphi )^{2}+e(\rho )\right),}$

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:

${\displaystyle e''={\frac {1}{\rho ))p',}$

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

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

${\displaystyle {\begin{aligned}{\frac {\partial \rho }{\partial t))&=+{\frac {\partial {\mathcal {H))}{\partial \varphi ))=-\nabla \cdot (\rho {\vec {u))),\\{\frac {\partial \varphi }{\partial t))&=-{\frac {\partial {\mathcal {H))}{\partial \rho ))=-{\frac {1}{2)){\vec {u))\cdot {\vec {u))-e',\end{aligned))}$

where ${\displaystyle {\vec {u))\ {\stackrel {\mathrm {def} }{=))\ \nabla \varphi }$ is the velocity and is vorticity-free. The second equation leads to the Euler equations:

${\displaystyle {\frac {\partial {\vec {u))}{\partial t))+({\vec {u))\cdot \nabla ){\vec {u))=-e''\nabla \rho =-{\frac {1}{\rho ))\nabla {p))$

after exploiting the fact that the vorticity is zero:

${\displaystyle \nabla \times {\vec {u))={\vec {0)).}$

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^[1][2]