{
  "id": "math/0610082",
  "version": "v1",
  "published": "2006-10-02T17:20:29.000Z",
  "updated": "2006-10-02T17:20:29.000Z",
  "title": "Geometry of a pair of second-order ODEs and Euclidean spaces",
  "authors": [
    "Richard Atkins"
  ],
  "comment": "20 pages",
  "journal": "Canad. Math. Bull. Vol. 49 (2), 2006 pp. 170-184",
  "categories": [
    "math.DG"
  ],
  "abstract": "This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler-Lagrange equations of the arclength action. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a pair of second-order ordinary differential equations be reparameterized so as to give, locally, the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second-order ODEs revealing the existence of 24 invariant functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-10-02T17:20:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "euclidean space",
      "second-order odes",
      "surface determines shortest paths",
      "second-order ordinary differential equations",
      "second-order differential equations"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10082A"
    }
  }
}