Note that this formalism only applies to nondissipative fluids.
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
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]