| Abstract: |
| The figure-eight choreography is the planar periodic solution of the equal mass 3-body problem in which three bodies chase each other in the same eight-shaped orbit. It is known that by changing the power $a$ of the interaction potential $-1/r^a$ between bodies the figure-eight choreography bifurcates to planar periodic solutions. The planar solution bifurcated at $a=0.9966$ is continued to $a=1$ and is called as Sim\`{o}`s H. In this talk, we show that Sim\`{o}`s H solution bifurcates at $a=0.8460$ to a nonplanar periodic solution composed of three distinct orbits, which has the same symmetry with the bifurcated solution found by Doedel {\it et al} by changing one mass in the figure-eight choreography. The locus of the orbit is symmetric in $\pi$ rotations in three perpendicular axes.
The symmetry of the figure-eight choreography is $D_6$ and by addition of the mirror symmetry in the plane of the motion, it is extended to $D_6 \times Z_2$ for motion in three dimensions. Thus, the group theoretic bifurcation theory based on action functional we developed can predicts possible bifurcations to nonplanar solutions from figure-eight choreography. Up to now, however, we have not found direct bifurcation to nonplanar solution from figure-eight choreography. |
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