Abstract
We generalize the well-known Hill's circular restricted three-body problem by assuming that the primary generates a Schwarzschild-type field of the form U = A/r + B/r3. The term in B influences the particle, but not the far secondary. Many concrete astronomical situations can be modelled via this problem. For the two-body problem primary-particle, a homoclinic orbit is proved to exist for a continuous range of parameters (the constants of energy and angular momentum, and the field parameter B > 0). Within the restricted three-body system, we prove that, under sufficiently small perturbations from the secondary, the homoclinic orbit persists, but its stable and unstable manifolds intersect transversely. Using a result of symbolic dynamics, this means the existence of a Smale horseshoe, hence chaotic behaviour. Moreover, we find that Hill's generalized problem (in our sense) is nonintegrable. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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