{
  "id": "2311.02770",
  "version": "v1",
  "published": "2023-11-05T21:00:14.000Z",
  "updated": "2023-11-05T21:00:14.000Z",
  "title": "Mass independent shapes for relative equilibria in the two dimensional positive curved three body problem",
  "authors": [
    "Toshiaki Fujiwara",
    "Ernesto Perez-Chavela"
  ],
  "comment": "21 pages, 2 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "In the planar three-body problem under Newtonian potential, it is well known that any masses, located at the vertices of an equilateral triangle generates a relative equilibrium, known as the Lagrange relative equilibrium. In fact, the equilateral triangle is the unique mass independent shape for a relative equilibrium in this problem. The two dimensional positive curved three-body problem, is a natural extension of the Newtonian three-body problem to the sphere $\\mathbb{S}^2$, where the masses are moving under the influence of the cotangent potential. S.~Zhu showed that in this problem, equilateral triangle on a rotating meridian can form a relative equilibria for any masses. This was the first report of mass independent shape on $\\mathbb{S}^2$ which can form a relative equilibrium. % In this paper, we show that, in addition to the equilateral triangle, three isosceles triangles on a rotating meridian, with $\\theta_{ij}=\\theta_{jk}=2^{-1}\\arccos((\\sqrt{2}-1)/2)$ for $(i,j,k)=(1,2,3)$, $(2,3,1)$, $(3,1,2)$ always form relative equilibria for any choice of the masses; where $\\theta_{ij}$ are the angles between the two masses $i$ and $j$ seen from the centre of $\\mathbb{S}^2$. Additionally we prove that, the equilateral and the above three isosceles relative equilibria are unique with this characteristic. We also prove that each relative equilibrium generated by a mass independent shape is not isolated from the other relative equilibria.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-05T21:00:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "70F07",
      "70F10",
      "70F15"
    ],
    "keywords": [
      "relative equilibrium",
      "equilateral triangle",
      "unique mass independent shape",
      "dimensional positive curved three-body problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}