On the isolating character of the "third" integral in a resonance case

Abstract

The problem of the transition of the third integral from the isolating to the nonisolating case is studied by numerical experiments using a potential field of the form V = -21A (x2+y2) - xy2. Particle orbits in this potential field have been computed for different energies and initial conditions. It is found that the third integral ceases to be isolating for a larger range of initial conditions with increasing total energy. However, even for energies higher than the energy of escape, there is a set of orbits that do not escape to infinity. Part of these orbits still indicate the existence of a very nearly isolating integral.