Note on the descriptions of the Euler-Poisson equations in various co-ordinate systems

In this note we derive the descriptions of the system of Euler-Poisson equations which governs the hydrodynamic evolution of gaseous stars in various co-ordinate systems. This note does not contain essentially new results for astrophysicists, but mathematically rigorous derivations cannot be found in the literatures written by physicists so that it will be useful to prepare details of rather stupidly honest derivations of the equations in various flames when we are going to push forward the mathematical research of the problem.