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

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

References in corpus (3)

• Linear Stability of Ring Systems
• On the dihedral n-body problem
• Symmetry-breaking for a restricted n-body problem in the Maxwell-ring configuration

Cited by in corpus (6)

• Choreographies in the $n$-vortex problem
• Relative periodic solutions of the n-vortex problem on the sphere
• Choreographies in the Discrete Nonlinear Schrödinger Equations
• Braids of the N-body problem by cabling a body in a central configuration
• From the Lagrange polygon to the figure eight I: Numerical evidence extending a conjecture of Marchal
• Comet and moon solutions in the time-dependent restricted $(n+1)$-body problem