Discovery of 10,059 new three-dimensional periodic orbits of general three-body problem

Abstract

A very few three-dimensional (3D) periodic orbits of general three-body problem (with three finite masses) have been discovered since Newton mentioned it in 1680s. Using a high-accuracy numerical strategy we discovered 10,059 three-dimensional periodic orbits of the three-body problem in the cases of $m_{1}=m_{2}=1$ and $m_{3}=0.1n$ where $1\leq n\leq 20$ is an integer, among which 1,996 (about 20\%) are linearly stable. Note that our approach is valid for arbitrary mass $m_{3}$ so that in theory we can gain an arbitrarily large amount of 3D periodic orbits of the three-body problem. In the case of three equal masses, we discovered twenty-one 3D ``choerographical'' periodic orbits whose three bodies move periodically in a single closed orbit. It is very interesting that, in the case of two equal masses, we discovered 273 three-dimensional periodic orbits with the two bodies ($m_{1}=m_{2}=1$) moving along a single closed orbit and the third ($m_{3}\neq 1$) along a different one: we name them ``piano-trio'' orbits, like a trio for two violins and one piano. To the best of our knowledge, all of these 3D periodic orbits have never been reported, indicating the novelty of this work. The large amount of these new 3D periodic orbits are helpful for us to have better understandings about chaotic properties of the famous three-body problem, which ``are, so to say, the only opening through which we can try to penetrate in a place which, up to now, was supposed to be inaccessible'', as pointed out by Poincar\'e, the founder of chaos theory.