Hamiltonian fluid mechanics

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

Irrotational barotropic flow[edit]

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

and the Hamiltonian by:

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:

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

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

where is the velocity and is vorticity-free. The second equation leads to the Euler equations:

after exploiting the fact that the vorticity is zero:

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]

See also[edit]