{
  "id": "2103.13004",
  "version": "v1",
  "published": "2021-03-24T06:17:57.000Z",
  "updated": "2021-03-24T06:17:57.000Z",
  "title": "On the $ C^{8/3} $-Regularisation of Simultaneous Binary Collisions in the Planar 4-Body Problem",
  "authors": [
    "Nathan Duignan",
    "Holger R. Dullin"
  ],
  "comment": "34 pages, 4 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "The dynamics of the 4-body problem allows for two binary collisions to occur simultaneously. It is known that in the collinear 4-body problem this simultaneous binary collision (SBC) can be block-regularised, but that the resulting block map is only $C^{8/3}$ differentiable. In this paper, it is proved that the $C^{8/3}$ differentiability persists for the SBC in the planar 4-body problem. The proof uses several geometric tools, namely, blow-up, normal forms, dynamics near normally hyperbolic manifolds of equilibrium points, and Dulac maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-24T06:17:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "70F16",
      "70F10"
    ],
    "keywords": [
      "simultaneous binary collision",
      "regularisation",
      "dulac maps",
      "equilibrium points",
      "normally hyperbolic manifolds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}