{
  "id": "1912.12375",
  "version": "v1",
  "published": "2019-12-28T00:41:41.000Z",
  "updated": "2019-12-28T00:41:41.000Z",
  "title": "Self-similarity in the Kepler-Heisenberg problem",
  "authors": [
    "Victor Dods",
    "Corey Shanbrom"
  ],
  "comment": "11 pages, 3 figures",
  "categories": [
    "math.DS",
    "cs.NA",
    "math-ph",
    "math.DG",
    "math.MP",
    "math.NA"
  ],
  "abstract": "The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. The dynamics are at least partially integrable, possessing two first integrals as well as a dilational momentum which is conserved by orbits with zero energy. The system is known to admit closed orbits of any rational rotation number, which all lie within the fundamental zero-energy integrable subsystem. Here we demonstrate that, under mild conditions, zero-energy orbits are self-similar. Consequently we find that these zero-energy orbits stratify into three families: future collision, past collision, and quasi-periodicity, with all collisions occurring in finite time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-28T00:41:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "70H12",
      "70F05",
      "53C17",
      "65P10"
    ],
    "keywords": [
      "kepler-heisenberg problem",
      "zero-energy orbits",
      "self-similarity",
      "three-dimensional sub-riemannian manifold",
      "rational rotation number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}