N-body choreographies on a p-limacon curve
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918005442609152 |
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| author | Fernandez-Guasti, Manuel Fujiwara, Toshiaki Perez-Chavela, Ernesto Zhu, Shuqiang |
| author_facet | Fernandez-Guasti, Manuel Fujiwara, Toshiaki Perez-Chavela, Ernesto Zhu, Shuqiang |
| contents | We consider an $N$--body problem under a harmonic potential of the form $\frac{1}{2}\sum κ_{jl} |q_j-q_l|^2$. A $p$-limaçon curve is a planar curve parametrized by $t$ given by $a(\cos t,\sin t)+b(\cos pt, \sin pt)$, where $a,b\in \mathbb{R}$, $p \in \mathbb{Z}$, and $t \in [0,2π]$. We study $N$-body choreographic motions constrained to a $p$-limaçon curve and establish necessary and sufficient conditions for their existence. Specifically, we prove that choreographic motions exist if and only if $p/N, (p \pm 1)/N \notin \mathbb{Z}$. Under an additional symmetry assumption on the force coefficients, we further refine these conditions. We also analyze the occurrence of collisions, showing that for given $p$ and $N$, at most $2(N-1)$ choices of $a/b$ lead to collisions. Furthermore, we find additional conserved quantities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21713 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | N-body choreographies on a p-limacon curve Fernandez-Guasti, Manuel Fujiwara, Toshiaki Perez-Chavela, Ernesto Zhu, Shuqiang Dynamical Systems We consider an $N$--body problem under a harmonic potential of the form $\frac{1}{2}\sum κ_{jl} |q_j-q_l|^2$. A $p$-limaçon curve is a planar curve parametrized by $t$ given by $a(\cos t,\sin t)+b(\cos pt, \sin pt)$, where $a,b\in \mathbb{R}$, $p \in \mathbb{Z}$, and $t \in [0,2π]$. We study $N$-body choreographic motions constrained to a $p$-limaçon curve and establish necessary and sufficient conditions for their existence. Specifically, we prove that choreographic motions exist if and only if $p/N, (p \pm 1)/N \notin \mathbb{Z}$. Under an additional symmetry assumption on the force coefficients, we further refine these conditions. We also analyze the occurrence of collisions, showing that for given $p$ and $N$, at most $2(N-1)$ choices of $a/b$ lead to collisions. Furthermore, we find additional conserved quantities. |
| title | N-body choreographies on a p-limacon curve |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2504.21713 |