Non-integrability of generalised Charlier and Saint-Germain problem

摘要

Certain generalisation of a classical problem of celestial mechanics is considered. A material point moves in the potential force field that is a superposition of a radial force and a constant force. The potential of the central forces is proportional to an integer power −n of the distance from the origin. For all positive integers n except n=1 non-integrability in meromorphic functions is proved. Case n=1 is integrable. For negative integer n analysis is much more complex: for n=−1 non-integrability is proved, for n=−2 additional first integral is found, for n=−3 although the system meets the necessary integrability conditions Poincaré cross-sections show presence of chaotic layers in phase space.