{
  "id": "0908.1631",
  "version": "v2",
  "published": "2009-08-12T07:59:11.000Z",
  "updated": "2010-04-22T05:45:57.000Z",
  "title": "Helmholtz conditions and symmetries for the time dependent case of the inverse problem of the calculus of variations",
  "authors": [
    "Ioan Bucataru",
    "Oana Constantinescu"
  ],
  "journal": "Journal of Geometry and Physics, vol. 60, no 11 (2010), 1710-1725",
  "doi": "10.1016/j.geomphys.2010.06.016",
  "categories": [
    "math.DG",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Fr\\\"olicher-Nijenhuis theory on the first jet bundle, $J^1\\pi$. We prove that a system of time dependent SODE, identified with a semispray $S$, is Lagrangian if and only if a special class, $\\Lambda^1_S(J^1\\pi)$, of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize the class $\\Lambda^1_S(J^1\\pi)$ of semi-basic 1-forms. Each such class contains the Poincar\\'e-Cartan 1-form of some Lagrangian function. We prove that if there exists a semi-basic 1-form in $\\Lambda^1_S(J^1\\pi)$, which is not a Poincar\\'e-Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-04-22T05:45:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58E30",
      "34A26",
      "70H03",
      "49N45"
    ],
    "keywords": [
      "time dependent case",
      "inverse problem",
      "variations",
      "second order ordinary differential equations",
      "global helmholtz conditions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Geometry and Physics",
      "year": 2010,
      "month": "Nov",
      "volume": 60,
      "number": 11,
      "pages": 1710
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010JGP....60.1710B"
    }
  }
}