Lagrangian Hydrodynamics, Entropy and Dissipation

The dynamics of dissipative fluids in Eulerian variables may be derived from an algebra of Leibniz brackets of observables, the metriplectic algebra, that extends the Poisson algebra of the frictionless limit of the sytem via a symmetric semidefinite component, that generates dissipative forces. The metriplectic algebra includes the conserved total Hamiltonian $H$, generating the non-dissipative part of dynamics, and the entropy S of those microscopic degrees of freedom draining energy irreversibly, that generates dissipation. This S is a Casimir invariant of the Poisson algebra to which the metriplectic algebra reduces in the frictionless limit. The role of S is as paramount as that of H, but this fact may be underestimated in the Eulerian formulation because S is not the only Casimir of the symplectic non-canonical part of the algebra. Instead, when the dynamics of the non-ideal fluid is written through the parcel variables of the Lagrangian formulation, the fact that entropy is symplectically invariant appears to be clearly related to its the microscopic degrees of freedom of the fluid, that do not participate at all to the symplectic canonical part of the algebra (which, indeed, involves and evolves only the macroscopic degrees of freedom of the fluid parcel).