Reflection symmetry in the equal-mass 1+N coorbital problem for 5≤ N≤8
Abstract
We study reflection symmetry in the equal-mass Newtonian 1+N coorbital problem for N=5,6,7,8. We prove that every collision-free solution of the angular central-configuration equations has a reflection axis. As a consequence, every non-coalescent limit of planar central configurations consisting of one dominant mass and N equal small masses is reflection-symmetric about a line through the dominant body. Our proof combines analytic arguments with rigorous computer-assisted verification. For each possible reflection, we introduce variables that measure the deviation from that reflection and vanish exactly when the configuration has that symmetry. The central-configuration equations yield a symmetric matrix equation in these variables. Concavity of the force derivative and estimates of the resulting quadratic forms reduce the symmetry problem to finitely many rigorously verified inequalities. Although the details depend on N, the same general method applies in all four cases. We also obtain, as a separate result, the complete three-class classification for N=5, up to the natural equivalences.
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BibTeX
@misc{huangfu2026reflection,
title = {{Reflection symmetry in the equal-mass $1+N$ coorbital problem for $5\leq N\leq8$}},
author = {Xiaoxiang Huangfu and Zijian Meng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31036},
eprint = {2610.31036},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31036},
}
RIS
TY - UNPB
TI - Reflection symmetry in the equal-mass 1+N coorbital problem for 5≤ N≤8
AU - Xiaoxiang Huangfu
AU - Zijian Meng
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31036
UR - https://arxiv.org/abs/2610.31036
AB - We study reflection symmetry in the equal-mass Newtonian 1+N coorbital problem for N=5,6,7,8. We prove that every collision-free solution of the angular central-configuration equations has a reflection axis. As a consequence, every non-coalescent limit of planar central configurations consisting of one dominant mass and N equal small masses is reflection-symmetric about a line through the dominant body. Our proof combines analytic arguments with rigorous computer-assisted verification. For each possible reflection, we introduce variables that measure the deviation from that reflection and vanish exactly when the configuration has that symmetry. The central-configuration equations yield a symmetric matrix equation in these variables. Concavity of the force derivative and estimates of the resulting quadratic forms reduce the symmetry problem to finitely many rigorously verified inequalities. Although the details depend on N, the same general method applies in all four cases. We also obtain, as a separate result, the complete three-class classification for N=5, up to the natural equivalences.
ER -