Mitsuru Shibayama

Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem

We study the isosceles three-body problem and show that there exist infinitely many families of relative periodic orbits converging to heteroclinic cycles between equilibria on the collision manifold in Devaneys blown-up coordinates. Towards this end, we prove that two types of heteroclinic orbits exist in much wider parameter ranges than previously detected, using self-validating interval arithmetic calculations, and we appeal to the previous results on heteroclinic orbits. Moreover, we give numerical computations for heteroclinic and relative periodic orbits to demonstrate our theoretical results. The numerical results also indicate that the two types of heteroclinic orbits and families of relative periodic orbits exist in wider parameter regions than detected in the theory and that some of them are related to Euler orbits.

Variational Proof of the Existence of the Super-Eight Orbit in the Four-Body Problem

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.

Multiple symmetric periodic solutions to the 2n-body problem with equal masses

Using the variational method, Chenciner and Montgomery (2000 Ann. Math. 152 881–901) proved the existence of an eight-shaped orbit of the planar three-body problem with equal masses. Since then a number of solutions to the N-body problem have been discovered. In particular, Ferrario and Terracini (2004 Invent. Math. 155 305–62) proved the existence of symmetric periodic solutions under a quite general setting. In this paper we consider the 2n-body problem with a certain symmetry and use the results of Ferrario and Terracini to prove the existence of multiple solutions for each n. Some of the solutions we find were already obtained by Chen (2003 Arch. Ration. Mech. Anal. 170 247–76) and Ferrario and Terracini (2004), but their argument does not allow us to distinguish the solutions obtained in this paper. By reducing the problem to the quotient space of the configuration space under the action of the group of symmetries and by observing boundary conditions of the solutions in the quotient space, we are able to distinguish these solutions and conclude that they are indeed distinct solutions. As a by-product, we can also determine the sign of the angular momentum of masses.