# Stable periodic orbits for three equal masses

For three equal masses in any number of dimensions (this might not be important, but 2D or 3D or 4D is fine) under just classical gravity (i.e., inverse-square force law), what stable periodic orbits are currently known?

The only solution I am aware of is the "figure-8" on Wikipedia, but surely there are more.

I think the biggest challenge here is stability. I try to define it below, but your definition is probably acceptable also. Anyway, I see hundreds of orbits posted on the internet, but I wonder if most of these degrade to chaos after continued simulation.

## Stability:

You provide a candidate $$t$$-parametrized "periodic limit orbit" $$O$$ (i.e., the functions $$r(t)$$ and $$v(t)$$) from which you choose any "generating point" $$G$$ (i.e., a specific $$t_0$$ prescribing $$r(t_0)$$ and $$v(t_0)$$).

Define the "epsilon-radius tube" about $$O$$ naturally. Define the "delta-radius ball" about $$G$$ a little more carefully by constraining points in this ball to not change the center of mass (otherwise, center of mass "drift" prevents any orbit from being stable).

Then, for every small epsilon I give, if you can provide a sufficiently-small delta such that the generated path stays within "a unitary transformation of the tube" forever, your candidate is stable.

EDIT: I changed from "the tube" to "a unitary transformation of the tube" to allow for a broader "rotating stability" (otherwise, rotation "drift" prevents any orbit from being stable). I believe this then covers all conservation laws. Also, I believe the 21 equal-mass families discovered here (and the 23 unequal-mass families discovered here) do meet my stability criteria.

• Since total energy and momentum must be conserved, I presume by "phase space" you mean a slice of phase space with, say, centre of mass $0$, momentum $0$, and constant energy. Commented Feb 28, 2022 at 5:14
• @RobertIsrael Thank you, good point, it is still a bit more complicated since angular momentum, at least, is also conserved. Anyway, I have edited my question to account for this detail. Commented Feb 28, 2022 at 16:50

The proposer gives as their definition of stability the standard notion of Lyapunov stability. Unfortunately, there are no known solutions for the planar or spatial three-body problem which are Lyapunov stable, the eight included. (One suspects none exist!) When people in celestial mechanics speak of an orbit they stumbled on being stable they almost always mean linearly stable. There are various other kinds, the implications amongst them being Nekhoroshev stability implies KAM stability implies linearly stable. Closely related to your question is what Hermann called the Oldest Problem in dynamical systems'' : arbitrarily close to any (relative) periodic orbit for the planar N-body problem is there an unbounded solution. (M. Herman, Some open problems in dynamical systems, Proc. Int. Congress of Math., Documenta Mathematica J. DMV, Extra volume ICM II, (1998) 797-808. ) A yes answer to the Oldest Problem would imply a response of no! no! no! to your search: no Lyapunov stable relative periodic orbits would exist. But note, the Oldest Problem is completely wide open.
A beautiful recent surprise however is that there do exist Lyapunov stable relative periodic orbits for the 3-body problem in Euclidean 4-space, using a 1/r potential. ( See Albouy, A. and Dullin R., 2020: Relative Equilibria of the 3-body Problem in $$\R^4$$ Journal of Geometric Mechanics, 12 (2020), 323-341 \url{http://www.aimsciences.org/article/doi/10.3934/jgm.2020012} \url{https://arxiv.org/abs/2002.00649}.)
As far as linearly stable solutions to the planar three body problem, there are almost certainly an infinite number of them. My guess is that for masses $$m_1 >> m_2 >> m_3$$ it is fairly straightforward to get an infinite number of KAM stable orbits using standard KAM techniques. But for three equal masses this has not been done, as best as I know.