Stellar dynamics in thin disk galaxies. I. A unified approach to hydrodynamic and orbit theories

We outline a procedure in stellar dynamics whereby we can evaluate closed moment equations as well as nonlinear stellar orbits. This formalism allows density wave disturbances which may be of finite amplitude. Furthermore, it facilitates close comparisons between the collective hydrodynamic behavior and the individual orbits of the stars.Here, the general analysis is presented within the context of nonlinear axisymmetric disturbances which are slowly varying in time and space. Stellar orbits are resolved into rapid epicyclic gyrations superposed on drifting guiding centers. This generalized epicyclic motion has an adiabatic invariant which determines its radial amplitude. Hydrodynamic equations also describe the collective stellar dynamic behavior. Closure of the equations occurs through a continuum form of the adiabatic invariant. To the lowest approximation, the hydrodynamic velocity equals the orbital drift velocity; the ratio of diagonal stresses is related to the epicyclic velocity ellipsoid; and there is a conservation theorem for mass and vorticity