A consistent hydrodynamic model is established to describe the evolution of thin disk galaxies. A previous result showing that the heat flow vanishes in the limit of small epicyclic motion is generalized and simplified, providing a closure relation for the hierarchy of fluid equations. Its systematic use at first allows the obtainment of a new invariant of the fluid motion, governing the evolution of the anisotropic pressure (stress) tensor. This might represent a step toward explicit adiabatic laws relating the pressure and the density of the stellar gas. The same closure relation is then used to derive a powerful variational principle, describing linearized perturbations of the system, which takes the pressure anisotropy fully into account. This principle is independent of the usual WKB (‘‘tightly wound’’) approximation, which is easily recovered in the asymptotic study of spiral waves and spiral modes.