Approximate action-angle variables for the figure-eight and periodic three-body orbits.

We use the maximally permutation-symmetric set of three-body coordinates that consist of the "hyper-radius" R=√[ρ(2)+λ(2)], the "rescaled area of the triangle" √[3]/2R(2) |ρ×λ|), and the (braiding) hyperangle Φ=arctan(2ρ·λ/λ(2)-ρ(2)) to analyze the "figure-eight" choreographic three-body motion discovered by Moore [Phys. Rev. Lett. 70, 3675 (1993)] in the Newtonian three-body problem. Here ρ,λ are the two Jacobi relative coordinate vectors. We show that the periodicity of this motion is closely related to the braiding hyperangle Φ. We construct an approximate integral of motion ̅G that together with the hyperangle Φ forms the action-angle pair of variables for this problem and show that it is the underlying cause of figure-eight motion's stability. We construct figure-eight orbits in two other attractive permutation-symmetric three-body potentials. We compare the figure-eight orbits in these three potentials and discuss their generic features, as well as their differences. We apply these variables to two new periodic, but nonchoreographic, orbits: One has a continuously rising Φ in time t, just like the figure-eight motion, but with a different, more complex, periodicity, whereas the other one has an oscillating Φ(t) temporal behavior.