{
  "abstract": "We study the spatial $(n+1)$-body problem formed by one heavy central mass together with $n$ equal masses placed on a single orbit of a polyhedral rotation group $H\\in\\{T,O,I\\}$, so that $n=|H|\\in\\{12,24,60\\}$. Imposing the symmetry $q_{L}=L\\,q_{I}$ for $L\\in H$ reduces the problem to a single $2π$-periodic reference curve, with reduced action $A_{H}=A_{0}+\\varepsilon A_{1}$, in which $\\varepsilon$ is the inverse central mass and $A_{0}$ is the Kepler action. At $\\varepsilon=0$ the critical set contains, as one connected component, the five-dimensional manifold of Kepler ellipses of minimal period $2π$, which we prove to be a nondegenerate critical manifold. A Lyapunov--Schmidt reduction along this manifold turns the continuation problem into the search for nondegenerate critical points of an explicit function $Φ(e,ψ)$ of the eccentricity $e$ and the spatial orientation $ψ$, a nondegeneracy we verify by a computer-assisted proof. We thereby obtain, for each of the three groups, families of periodic solutions of the $(n+1)$-body problem with $n+1\\in\\{13,25,61\\}$ bodies, bifurcating from Kepler ellipses and carrying the full tetrahedral, octahedral, or icosahedral symmetry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10242",
  "author_total": 3,
  "authors": [
    {
      "name": "Kevin Constantineau"
    },
    {
      "name": "Carlos García-Azpeitia"
    },
    {
      "name": "Jean-Philippe Lessard"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bh2ekx6be1fpn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10242",
  "fulltext_url": null,
  "id": "bh2ekx6be1fpn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10242",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7212113727",
  "pdf_url": "https://arxiv.org/pdf/2609.10242",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bh2ekx6be1fpn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.10242",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10242"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Platonic constellations of periodic motions in the $(n + 1)$-body problem",
  "title_text": "Platonic constellations of periodic motions in the (n + 1)-body problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/bh2ekx6be1fpn",
  "venue": "arXiv",
  "versions": []
}