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A new geometric method and its applications in the N-body problem

Abstract

In 2000, Chenciner and Montgomery proved the existence of the figure-eight solution in the planar three-body problem with equal masses by using the variational method. Since then, a number of new periodic solutions have been discovered and proven to exist. A workshop on Variational Methods in Celestial Mechanics was organized by Chenciner and Montgomery in 2003 to address the possible applications of variational method in studying the Newtonian N-body problem, while several open problems were proposed by the attending experts. The existence of the Broucke-Henon orbit is one of these open problems, which was proposed by Venturelli.

By introducing a new geometric argument, we show that under an appropriate topological constraint, the action minimizer must be either the Schubart orbit or the Broucke-Henon orbit. This geometric argument can be applied to many orbits in the three-body and four-body problems. The report is based on joint works with Wentian Kuang, Tiancheng Ouyang, Zhifu Xie and Rongchang Liu.