{
  "id": "2204.01420",
  "version": "v1",
  "published": "2022-04-04T12:14:36.000Z",
  "updated": "2022-04-04T12:14:36.000Z",
  "title": "Braids, metallic ratios and periodic solutions of the $2n$-body problem",
  "authors": [
    "Yuika Kajihara",
    "Eiko Kin",
    "Mitsuru Shibayama"
  ],
  "comment": "25 pages, many figures",
  "categories": [
    "math.DS",
    "math.GT"
  ],
  "abstract": "Periodic solutions of the planar $N$-body problem determine braids through the trajectory of $N$ bodies. Braid types can be used to classify periodic solutions. According to the Nielsen-Thurston classification of surface automorphisms, braids fall into three types: periodic, reducible and pseudo-Anosov. To a braid of pseudo-Anosov type, there is an associated stretch factor greater than 1, and this is a conjugacy invariant of braids. In 2006, the third author discovered a family of multiple choreographic solutions of the planar $2n$-body problem. We prove that braids obtained from the solutions in the family are of pseudo-Anosov type, and their stretch factors are expressed in metallic ratios. New numerical periodic solutions of the planar $2n$-body problem are also provided.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-04T12:14:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K10",
      "57K20",
      "70F10"
    ],
    "keywords": [
      "periodic solutions",
      "metallic ratios",
      "body problem determine braids",
      "pseudo-anosov type",
      "multiple choreographic solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}