Many astronomical systems require for their description in the frame of Einstein's theory of gravity not just one but several reference systems. In the first post-Newtonian approximation the Damour-Soffel-Xu (DSX) formalism presents a new and improved treatment of celestial mechanics and astronomical reference systems for the gravitational N-body problem. In the DSX-formalism the astronomical bodies are characterized by their Blanchet-Damour (BD) mass- and spin-multipole moments. However, the time dependence of these moments requires additional dynamical equations, usually local flow equations describing the internal motions inside the bodies or additional assumptions about them. In this article the internal motion of astronomical bodies will be adressed within the 1st post-Newtonian approximation to Einstein's theory of gravity. A concept of quasi-rigid bodies will be introduced; after that, astronomical fluid and elastic bodies will be discussed.
