Transition from regular motion to chaos in a logarithmic potential

We investigate the properties of motion in a logarithmic galactic potential. The model can be considered to describe the motion in the meridian, $r-z$ plane, of an elliptical galaxy with a dense nucleus or bulge of radius c. For a given value of c, there is a critical value of the angular momentum Lzc such as for $L_z \leq L_{zc}$, stars, moving near the galactic plane, are scattered to higher scale z heights displaying chaotic motion. Our numerical calculations show that there exists a linear relationship between the radius of the nucleus and the critical value of the angular momentum. This linear relationship can be found using some elementary theoretical arguments. We use the distribution of radial velocities in order to distinguish ordered from chaotic motion. Comparison with previous work is also made