{
  "id": "math/0007059",
  "version": "v1",
  "published": "2000-07-10T19:17:28.000Z",
  "updated": "2000-07-10T19:17:28.000Z",
  "title": "From flows and metrics to dynamics",
  "authors": [
    "C. Udriste",
    "A. Udriste"
  ],
  "comment": "12 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Recall that a vector field on an n-dimensional differentiable manifold M is a mapping X defined on M with values in the tangent bundle TM that assigns to each point $x\\in M$ a vector X(x) in the tangent space $T_x M$. A vector field may be interpreted alternatively as the right-hand side of an autonomous system of first-order ordinary differential equations, i.e., a flow. Now we show that any flow can be enveloped by a conservative dynamics using a semi-Riemann metric g on M. This kind of dynamics was called {\\it geometric dynamics} [7]-[9]. The given vector field, the initial semi-Riemann metric, the Levi-Civita connection, and an associated (1,1)-tensor field are used to build a new geometric structure (e.g., semi-Riemann-Jacobi, semi-Riemann-Jacobi-Lagrange, semi-Finsler-Jacobi, etc) on the manifold M ensuring that all the trajectories of a geometric dynamics are pregeodesics (Lorentz-Udri\\c{s}te world-force law). Implicitly, we solved a problem rised first by Poincar\\'e: find a suitable geometric structure that converts the trajectories of a given vector field into geodesics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-07-10T19:17:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "70G45",
      "70S05",
      "53C50"
    ],
    "keywords": [
      "vector field",
      "geometric structure",
      "geometric dynamics",
      "first-order ordinary differential equations",
      "tangent bundle tm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......7059U"
    }
  }
}