{
  "id": "1511.00070",
  "version": "v1",
  "published": "2015-10-31T04:02:33.000Z",
  "updated": "2015-10-31T04:02:33.000Z",
  "title": "The linear stability of elliptic Euler-Moulton solutions of the n-body problem via those of 3-body problems",
  "authors": [
    "Qinglong Zhou",
    "Yiming Long"
  ],
  "comment": "21 pages",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper, we study the linear stability of the elliptic collinear solutions of the classical $n$-body problem, where the $n$ bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler-Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler-Moulton collinear solution of $n$-bodies splits into $(n-1)$ independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler $2$-body problem at Kepler elliptic orbit, and each of the other $(n-2)$ systems is the essential part of some linearized Hamiltonian system at an elliptic Euler collinear solution of the $3$-body problem whose mass parameter is modified. Then using analytical results proved by Zhou and Long in [21] and by Hu and Ou in [5] on $3$-body Euler solutions, the linear stability of such a solution in the $n$-body problem is reduced to those of the corresponding elliptic Euler collinear solutions in the $3$-body problems. As an example, we carried out the detailed derivation of the linear stability for an elliptic Euler-Moulton solution of the $4$-body problem with two small masses in the middle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-31T04:02:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58E05",
      "37J45",
      "34C25"
    ],
    "keywords": [
      "elliptic euler-moulton solution",
      "linear stability",
      "elliptic euler collinear solution",
      "linearized hamiltonian system",
      "n-body problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}