On the four-body limaçon choreography: maximal superintegrability and choreographic fragmentation
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911043917185024 |
|---|---|
| author | Escobar-Ruiz, Adrian M Fernandez-Guasti, Manuel |
| author_facet | Escobar-Ruiz, Adrian M Fernandez-Guasti, Manuel |
| contents | In this paper, as a continuation of [Fernandez-Guasti, \textit{Celest Mech Dyn Astron} 137, 4 (2025)], we demonstrate the maximal superintegrability of the reduced Hamiltonian, which governs the four-body choreographic planar motion along the trisectrix limaçon (resembling a folded figure eight), in the six-dimensional space of relative motion. The pairwise interaction potential $V(r_{ij})$ among the four bodies is a quadratic expression in the relative distances $r_{ij}$, with a combination of positive and negative coefficients. The corresponding eleven integrals of motion in the Liouville-Arnold sense are presented explicitly. Specifically, it is shown that the reduced Hamiltonian admits complete separation of variables in Jacobi-like variables. The emergence of this choreography is not a direct consequence of maximal superintegrability. Rather, it originates from the existence of \textit{particular integrals} and the phenomenon of \textit{particular involution}. We also provide a detailed analysis of the fragmentation of a general four-body choreographic motion into two isomorphic two-body choreographies, as well as the reverse process, namely, the fusion of two-body choreographies into a four-body configuration. This model combines choreographic motion with maximal superintegrability, a seldom-studied interplay in classical mechanics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16403 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the four-body limaçon choreography: maximal superintegrability and choreographic fragmentation Escobar-Ruiz, Adrian M Fernandez-Guasti, Manuel Mathematical Physics In this paper, as a continuation of [Fernandez-Guasti, \textit{Celest Mech Dyn Astron} 137, 4 (2025)], we demonstrate the maximal superintegrability of the reduced Hamiltonian, which governs the four-body choreographic planar motion along the trisectrix limaçon (resembling a folded figure eight), in the six-dimensional space of relative motion. The pairwise interaction potential $V(r_{ij})$ among the four bodies is a quadratic expression in the relative distances $r_{ij}$, with a combination of positive and negative coefficients. The corresponding eleven integrals of motion in the Liouville-Arnold sense are presented explicitly. Specifically, it is shown that the reduced Hamiltonian admits complete separation of variables in Jacobi-like variables. The emergence of this choreography is not a direct consequence of maximal superintegrability. Rather, it originates from the existence of \textit{particular integrals} and the phenomenon of \textit{particular involution}. We also provide a detailed analysis of the fragmentation of a general four-body choreographic motion into two isomorphic two-body choreographies, as well as the reverse process, namely, the fusion of two-body choreographies into a four-body configuration. This model combines choreographic motion with maximal superintegrability, a seldom-studied interplay in classical mechanics. |
| title | On the four-body limaçon choreography: maximal superintegrability and choreographic fragmentation |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2504.16403 |