Some properties of an intrinsically degenerate Hamiltonian system

Using theoretical arguments we study some properties of a time independent, two-dimensional Hamiltonian system in the 1:1 resonance case. In particular we find the frequency of the oscillation near the elliptic points, the ratio of the semiaxes of the ellipses surrounding them and the inclination angle of the asymptotes of the hyperbolic orbits in the original variablesx–p _x. The results are in satisfactory agreement with those given by numerical integration of the equations of motion. Some possible applications for the motion of stars in elliptical galaxies are also discussed.