Abstract
Using the variational method, Chenciner and Montgomery (Ann Math 152:881–901, 2000) proved the existence of an eight-shaped periodic solution of the planar three-body problem with equal masses. Just after the discovery, Gerver 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 scaling technique established by Tanaka (Ann Inst H Poincaré Anal Non Linéaire 10:215–238, 1993), (Proc Am Math Soc 122:275–284, 1994) and investigate the asymptotic behavior of a binary collision.
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Communicated by P. Rabinowitz
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Shibayama, M. Variational Proof of the Existence of the Super-Eight Orbit in the Four-Body Problem. Arch Rational Mech Anal 214, 77–98 (2014). https://doi.org/10.1007/s00205-014-0753-x
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DOI: https://doi.org/10.1007/s00205-014-0753-x