| Abstract: |
| We found a relationship between figure-eight choreography for the equal mass three-body problem under homogeneous potential $-1/r^a$ and H solution found by Sim\`{o} through Morse index $N(a)$. The H solution is periodic and consists of three different eight-shaped orbits symmetric in both x and y axes. It coincides with figure-eight choreography at $a=0.9966$ where $N(a)$ changes. In general, changes of Morse index predict several periodic solutions close to figure-eight choreography. At $a=1.3424$, $N(a)$ changes and a periodic solution H`, less symmetric than the H, symmetric only in y axis, is suggested. For the system under Lenard-Jones-type (LJ) potential $1/r^{12}-1/r^6$, the figure-eight choreography with period $T$ is not scalable. Changes of its Morse index in $T$ predicts another type of periodic solution 8`, an eight-shaped choreography not symmetric either in x or y axis but at origin. Numerically the H solution can be found with an isosceles triangle configuration at $t=0$ and Euler configuration at $t=T/4$, the H` with incongruent isosceles triangle configurations at $t=0$ and $t=T/2$, and 8` with incongruent Euler configurations at $t=0$ and $t=T/6$. We found numerically the H and the 8` solution under LJ but did not find the H` either under homogeneous or LJ. |
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