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arXiv · 1702.03990

Symmetries and choreographies in families bifurcating from the polygonal relative equilibrium of the $n$-body problem

Renato Calleja·Eusebius Doedel·Carlos García-Azpeitia
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Abstract

We use numerical continuation and bifurcation techniques in a boundary value setting to follow Lyapunov families of periodic orbits. These arise from the polygonal system of $n$ bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, correspond to choreographies. We present a sample of the many choreographies that we have determined numerically along the Lyapunov families and along bifurcating families, namely for the cases $n=4,~6,~7,~8$, and $9$. We also present numerical results for the case where there is a central body that affects the choreography, but that does not participate in it. Animations of the families and the choreographies can be seen at the link: http://mym.iimas.unam.mx/renato/choreographies/index.html

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Renato Calleja, Eusebius Doedel, Carlos García-Azpeitia. 2017-02-13. Symmetries and choreographies in families bifurcating from the polygonal relative equilibrium of the $n$-body problem. https://doi.org/10.1007/s10569-018-9841-9

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