Linear stability of the elliptic relative equilibria for the restricted N-body problem: two special cases
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866912038459015168 |
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| author | Xie, Jiashengliang Liu, Bowen Zhou, Qinglong |
| author_facet | Xie, Jiashengliang Liu, Bowen Zhou, Qinglong |
| contents | In this paper, we consider the elliptic relative equilibria of the restricted $N$-body problems, where the $N-1$ primaries form an Euler-Moulton collinear central configuration or a $(1+n)$-gon central configuration. We obtain the symplectic reduction to the general restricted $N$-body problem. For the first case, by analyzing the relationship between this restricted $N$-body problems and the elliptic Lagrangian solutions, we obtain the linear stability of the restricted $N$-body problem by the $ω$-Maslov index. Via numerical computations, we also obtain conditions of the stability on the mass parameters under $N=4$ and the symmetry of the central configuration. For the second case, there exist three positions $S_1,S_2$ and $S_3$ of the massless body (up to rotations of angle $\frac{2π}{n}$). For ${m_0\over m}$ sufficiently large, we show that the elliptic relative equilibria is linearly unstable if the eccentricity $0\le e<e_0$ and the massless body lies at $S_1$ or $S_2$; while the elliptic relative equilibria is linear stability if the massless body lies at $S_3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_00286 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Linear stability of the elliptic relative equilibria for the restricted N-body problem: two special cases Xie, Jiashengliang Liu, Bowen Zhou, Qinglong Dynamical Systems Mathematical Physics 70F10, 70H14, 34C25 In this paper, we consider the elliptic relative equilibria of the restricted $N$-body problems, where the $N-1$ primaries form an Euler-Moulton collinear central configuration or a $(1+n)$-gon central configuration. We obtain the symplectic reduction to the general restricted $N$-body problem. For the first case, by analyzing the relationship between this restricted $N$-body problems and the elliptic Lagrangian solutions, we obtain the linear stability of the restricted $N$-body problem by the $ω$-Maslov index. Via numerical computations, we also obtain conditions of the stability on the mass parameters under $N=4$ and the symmetry of the central configuration. For the second case, there exist three positions $S_1,S_2$ and $S_3$ of the massless body (up to rotations of angle $\frac{2π}{n}$). For ${m_0\over m}$ sufficiently large, we show that the elliptic relative equilibria is linearly unstable if the eccentricity $0\le e<e_0$ and the massless body lies at $S_1$ or $S_2$; while the elliptic relative equilibria is linear stability if the massless body lies at $S_3$. |
| title | Linear stability of the elliptic relative equilibria for the restricted N-body problem: two special cases |
| topic | Dynamical Systems Mathematical Physics 70F10, 70H14, 34C25 |
| url | https://arxiv.org/abs/2310.00286 |