{
  "id": "math/0105082",
  "version": "v2",
  "published": "2001-05-10T17:44:46.000Z",
  "updated": "2002-10-03T17:08:22.000Z",
  "title": "Morse theory on spaces of braids and Lagrangian dynamics",
  "authors": [
    "R. W. Ghrist",
    "J. B. Van den Berg",
    "R. C. Vandervorst"
  ],
  "comment": "Revised version: numerous changes in exposition. Slight modification of two proofs and one definition; 55 pages, 20 figures",
  "categories": [
    "math.DS",
    "math-ph",
    "math.GT",
    "math.MP",
    "math.SG"
  ],
  "abstract": "In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index and provides a gloablization of `lap number' techniques used in scalar parabolic PDEs. In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-10-03T17:08:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37J45",
      "37B30",
      "57M25"
    ],
    "keywords": [
      "morse theory",
      "lagrangian dynamics",
      "parabolic flows",
      "evolve singular braid diagrams",
      "topological invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......5082G"
    }
  }
}