{
  "id": "2205.10514",
  "version": "v1",
  "published": "2022-05-21T06:34:32.000Z",
  "updated": "2022-05-21T06:34:32.000Z",
  "title": "Linear stability of the elliptic relative equilibria for the restricted $4$-body problem: the Euler case",
  "authors": [
    "Bowen Liu",
    "Qinglong Zhou"
  ],
  "comment": "23 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1907.13475; substantial text overlap with arXiv:1206.6162 by other authors",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper, we consider the elliptic relative equilibria of the restricted $4$-body problems, where the three primaries form an Euler collinear configuration and the four bodies span $\\mathbf{R}^2$. We obtain the symplectic reduction to the general restricted $N$-body problem. By analyzing the relationship between this restricted $4$-body problems and the elliptic Lagrangian solutions, we obtain the linear stability of the restricted $4$-body problem by the $\\omega$-Maslov index. Via numerical computations, we also obtain conditions of the stability on the mass parameters for the symmetric cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-21T06:34:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "70F10",
      "70H14",
      "34C25"
    ],
    "keywords": [
      "body problem",
      "elliptic relative equilibria",
      "linear stability",
      "euler case",
      "euler collinear configuration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}