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

Abstract

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" $\frac{\sqrt{3}}{2R^2}|\mathbf{\rho }\times \mathbf{\lambda }|$) and the (braiding) hyper-angle $\varphi =\arctan (\frac{2\mathbf{\rho }\cdot \mathbf{\lambda }}{{\lambda }^2-{\rho }^2})$, to analyze the "figure-eight" choreographic three-body motion discovered by Moore \cite{Moore1993} in the Newtonian three-body problem. Here $\mathbf{\rho },\mathbf{\lambda }$ are the two Jacobi relative coordinate vectors. We show that the periodicity of this motion is closely related to the braiding hyper-angle $\varphi$. We construct an approximate integral of motion $\bar{G}$ that together with the hyper-angle $\varphi$ 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 non-choreographic 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.

11 pages, 19 figures

References in corpus (7)

• Detailed Analysis of the Three Quark Potential in SU(3) Lattice QCD
• Choreographic solution to the general relativistic three-body problem
• Choreographic Three Bodies on the Lemniscate
• Synchronised Similar Triangles for Three-Body Orbit with Zero Angular Momentum
• On the smooth cross-over transition from the Delta-string to the Y-string three-quark potential
• Low-lying spectrum of the Y-string three-quark potential using hyper-spherical coordinates
• Evolution of the Moment of Inertia of Three-Body Figure-Eight Choreography

Cited by in corpus (2)

• Three Classes of Newtonian Three-Body Planar Periodic Orbits
• Periodic three-body orbits with vanishing angular momentum in the Jacobi-Poincare "strong" potential