Relative equilibria, linear stability and electromagnetic curvature
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914461305012224 |
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| author | Asselle, Luca Testolina, Giorgia |
| author_facet | Asselle, Luca Testolina, Giorgia |
| contents | In this paper we study the linear stability of relative equilibria in the Newtonian $n$-body problem from the viewpoint of electromagnetic systems. We first examine the effect of the ambient dimension on stability, starting from the Lagrange equilateral triangle solutions of the three-body problem in $\mathbb R^4$. We then initiate a new approach to stability based on electromagnetic curvature. In a two-dimensional model, we relate linear stability to both the Mañé critical value and to the behavior of the zero set of the electromagnetic curvature, highlighting a change in its topology at the stability threshold. This criterion is then applied to the planar $n$-body problem: in the three-body case, we recover Routh's classical criterion, and, more generally, we obtain an instability criterion for relative equilibria whose reduced linearized dynamics splits along invariant symplectic planes. These results suggest a new geometric perspective on linear stability and on questions related to Moeckel's conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_07975 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relative equilibria, linear stability and electromagnetic curvature Asselle, Luca Testolina, Giorgia Dynamical Systems In this paper we study the linear stability of relative equilibria in the Newtonian $n$-body problem from the viewpoint of electromagnetic systems. We first examine the effect of the ambient dimension on stability, starting from the Lagrange equilateral triangle solutions of the three-body problem in $\mathbb R^4$. We then initiate a new approach to stability based on electromagnetic curvature. In a two-dimensional model, we relate linear stability to both the Mañé critical value and to the behavior of the zero set of the electromagnetic curvature, highlighting a change in its topology at the stability threshold. This criterion is then applied to the planar $n$-body problem: in the three-body case, we recover Routh's classical criterion, and, more generally, we obtain an instability criterion for relative equilibria whose reduced linearized dynamics splits along invariant symplectic planes. These results suggest a new geometric perspective on linear stability and on questions related to Moeckel's conjecture. |
| title | Relative equilibria, linear stability and electromagnetic curvature |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2604.07975 |