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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2511.03134 |
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| _version_ | 1866912721742594048 |
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| author | Sánchez-Cerritos, Juan Manuel Torres-Hernández, Mayte |
| author_facet | Sánchez-Cerritos, Juan Manuel Torres-Hernández, Mayte |
| contents | We present a variational approach to obtain periodic solutions of the $N$-body problem, in particular the 'figure-eight' solution for three equal masses. The central idea is to explicitly optimize the \emph{spatial scale} within the Lagrangian action, leading to the functional $\mathcal F = K^{α/(α+2)} V^{2/(α+2)}$. We prove the existence of critical points of $\mathcal F$ that enforce a curve with a single self-crossing, and show that every reparametrized critical curve satisfies Newton's equations and is free of collisions. This framework recovers the Chenciner-Montgomery 'eight' (for $α=1$) and extends to the whole range $0<α<2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03134 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Variational Approach to Planar Choreographies via Ekeland's Principle Sánchez-Cerritos, Juan Manuel Torres-Hernández, Mayte Dynamical Systems We present a variational approach to obtain periodic solutions of the $N$-body problem, in particular the 'figure-eight' solution for three equal masses. The central idea is to explicitly optimize the \emph{spatial scale} within the Lagrangian action, leading to the functional $\mathcal F = K^{α/(α+2)} V^{2/(α+2)}$. We prove the existence of critical points of $\mathcal F$ that enforce a curve with a single self-crossing, and show that every reparametrized critical curve satisfies Newton's equations and is free of collisions. This framework recovers the Chenciner-Montgomery 'eight' (for $α=1$) and extends to the whole range $0<α<2$. |
| title | A Variational Approach to Planar Choreographies via Ekeland's Principle |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2511.03134 |