Morse index for figure-eight choreographies of the planar equal mass three-body problem

Abstract

We report on numerical calculations of Morse index for figure-eight choreographic solutions to a system of three identical bodies in a plane interacting through homogeneous potential, $-1/r^a$, or through Lennard-Jones-type (LJ) potential, $1/r^{12} - 1/r^6$, where $r$ is a distance between the bodies. The Morse index is a number of independent variational functions giving negative second variation $S^{(2)}$ of action functional $S$. We calculated three kinds of Morse indices, $N$, $N_c$ and $N_e$, in the domain of the periodic, the choreographic and the figure-eight choreographic function, respectively. For homogeneous system, we obtain $N=4$ for $0 \le a < a_0$, $N=2$ for $a_0 < a < a_1$, $N=0$ for $a_1 < a$, and $N_c=N_e=0$ for $0 \le a$, where $a_0=0.9970$ and $a_1=1.3424$. For $a=1$, we show a strong relationship between the figure-eight choreography and the periodic solution found by Sim\'{o} through the $S^{(2)}$. For LJ system, we calculated the index for the solution tending to the figure-eight solution of $a=6$ homogeneous system for the period $T \to \infty$. We obtain $N$, $N_c$ and $N_e$ as monotonically increasing functions of the gradual change in $T$ from $T \to \infty$, which start with $N=N_c=N_e=0$, jump at the smallest $T$ by $1$, and reach $N=12$, $N_c=4$, and $N_e=1$ for $T \to \infty$ in the other branch.