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
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
