Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Morse index for figure-eight choreographies of the planar equal mass
  three-body problem   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.

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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
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The categories of this paper are: funct-an
33
The categories of this paper are: plasm-ph
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The categories of this paper are: cs
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The categories of this paper are: cond-mat
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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: adap-org