Variational proof of the existence of the super-eight orbit in the 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).