Hamiltonian fluid mechanics

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

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]

• Luke's variational principle
• Hamiltonian field theory

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