Variational proof of the existence of the super-eight orbit in the four-body problem

TL;DR

This paper proves the existence of a super-eight periodic orbit in the planar four-body problem with equal masses using variational methods, extending techniques previously applied to the three-body problem.

Contribution

It provides a rigorous variational proof for the super-eight orbit in the four-body problem, addressing collision elimination with established mathematical techniques.

Findings

Existence of super-eight orbit in four-body problem proven.

Application of Tanaka's collision elimination technique.

Extension of variational methods from three- to four-body problem.

Abstract

Using the variational method, Chenciner and Montgomery (2000 Ann. Math. 152 881--901) proved the existence of an eight-shaped periodic solution of the planar three-body problem with equal masses. Just after the discovery, Gerver have numerically found a similar periodic solution called "super-eight" in the planar four-body problem with equal mass. In this paper we prove the existence of the super-eight orbit by using the variational method. The difficulty of the proof is to eliminate the possibility of collisions. In order to solve it, we apply the technique established by Tanaka (1993 Ann. Inst. H. Poincar'e Anal. Non Lin'eaire 10, 215--238, 1994 Proc. Amer. Math. Soc. 122, 275--284).