Quantitative Linear Stability Analysis of Elliptic Relative Equilibria in the Planar N-Body Problem
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914033081253888 |
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| author | Hu, Xijun Ou, Yuwei Sun, Jiexin |
| author_facet | Hu, Xijun Ou, Yuwei Sun, Jiexin |
| contents | An elliptic relative equilibrium (ERE) is a special solution of the planar $N$-body problem generated by a central configuration. Its linear stability depends on the eccentricity $e$ and the masses of the bodies. However, for $e>0$, the variational equations become non-autonomous and highly complex, particularly near $e=1$, where the system exhibits a singularity. This complicates the stability analysis as $e$ approaches one, making it challenging to derive a rigorous quantitative estimate for the stable region across $e\in[0,1)$. In this work, we address this problem. Using trace formulas for the non-degenerate Hamiltonian system of EREs, we establish an upper bound ensuring non-degeneracy for all $e\in[0,1)$. As key applications, we provide explicit stability estimates for the Lagrange, Euler, and regular $(1+n)$-gon EREs over the full range of eccentricity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09809 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative Linear Stability Analysis of Elliptic Relative Equilibria in the Planar N-Body Problem Hu, Xijun Ou, Yuwei Sun, Jiexin Dynamical Systems An elliptic relative equilibrium (ERE) is a special solution of the planar $N$-body problem generated by a central configuration. Its linear stability depends on the eccentricity $e$ and the masses of the bodies. However, for $e>0$, the variational equations become non-autonomous and highly complex, particularly near $e=1$, where the system exhibits a singularity. This complicates the stability analysis as $e$ approaches one, making it challenging to derive a rigorous quantitative estimate for the stable region across $e\in[0,1)$. In this work, we address this problem. Using trace formulas for the non-degenerate Hamiltonian system of EREs, we establish an upper bound ensuring non-degeneracy for all $e\in[0,1)$. As key applications, we provide explicit stability estimates for the Lagrange, Euler, and regular $(1+n)$-gon EREs over the full range of eccentricity. |
| title | Quantitative Linear Stability Analysis of Elliptic Relative Equilibria in the Planar N-Body Problem |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2509.09809 |