Generalizing the restricted three-body problem. The Bianular and Tricircular coherent problems

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain e-mail: angel@maia.ub.es

Corresponding author: F. Gabern, gabern@mat.ub.es

Received: 3 December 2003
Accepted: 4 March 2004

Abstract

In this paper we construct two models for the motion of a particle under the gravitational attraction of Sun, Jupiter, Saturn and Uranus, that can be seen as a generalization of the well known Restricted Three-Body Problem (RTBP). Both models are obtained by computing quasi-periodic solutions – with two basic frequencies – of a suitable N-body problem. The first model is based on a quasi-periodic solution of the planar Sun-Jupiter-Saturn Three-Body problem, that tries to approach the real motion of Jupiter. The second model is based on a quasi-periodic solution of the Sun-Jupiter-Saturn-Uranus Four-Body problem. In both cases, we derive the equations of motion for a particle under the gravitational attraction of these bodies as a quasi-periodic time-dependent perturbation of the well-known RTBP.