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Integrability of Motions in Galactic Potentials

Abstract The dynamics exhibited by systems, such as galaxies, are dominated by the isolating integrals of the motion. The most common are the energy and angular momentum integrals. The motions in a system with a full complement of isolating integrals are regular, that is, periodic or quasi-periodic. Such a system is integrable. If there is a deficiency in the number of integrals, then the motions are chaotic. There is a fundamental quantative difference in the motion, depending on the number of integrals. A technique, called Generalised Painlevé analysis, based on complex variable theory allows the user to determine if a system is integrable. Two new integrable cases of the Henon-Heiles system are presented, bringing the total number of such integrable potentials to five. It is highly probable that there are no further integrable cases of the Henon-Heiles potential. Five cases of the quartic Verhulst potential, defined by certain restrictions on the coefficients, which are found to be integrable are summarised.