 # Hamiltonian fluid mechanics

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

{ φ ( x ) , ρ ( y ) } = δ d ( x y )

and the Hamiltonian by:

H = d d x H = d d x ( 1 2 ρ ( φ ) 2 + e ( ρ ) ) ,

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 = 1 ρ p ,

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

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

ρ t = + H φ = ( ρ u ) , φ t = H ρ = 1 2 u u e ,

where u   = d e f   φ is the velocity and is vorticity-free. The second equation leads to the Euler equations:

u t + ( u ) u = e ρ = 1 ρ p

after exploiting the fact that the vorticity is zero:

× u = 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

## References

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