We use the maximally permutation-symmetric set of three-body coordinates that consist of the “hyper-radius” $R=\sqrt{{\rho }^2+{\lambda }^2}$, the “rescaled area of the triangle” $\frac{\sqrt{3}}{2R^2}|\mathit{\rho }\times \mathit{\lambda }|$), and the (braiding) hyperangle $\varphi =\arctan (\frac{2\rho ·\lambda }{{\lambda }^2-{\rho }^2})$ 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 $\mathit{\rho },\mathit{\lambda }$ are the two Jacobi relative coordinate vectors. We show that the periodicity of this motion is closely related to the braiding hyperangle $\varphi$. We construct an approximate integral of motion $\stackrel{̲}{G}$ that together with the hyperangle $\varphi$ 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 $\varphi$ in time $t$, just like the figure-eight motion, but with a different, more complex, periodicity, whereas the other one has an oscillating $\varphi (t)$ temporal behavior.

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