{
  "id": "2307.07883",
  "version": "v1",
  "published": "2023-07-15T21:01:50.000Z",
  "updated": "2023-07-15T21:01:50.000Z",
  "title": "Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge",
  "authors": [
    "Erasmo Caponio",
    "Dario Corona",
    "Roberto Giambò",
    "Paolo Piccione"
  ],
  "comment": "29 pages",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider an autonomous, indefinite Lagrangian $L$ admitting an infinitesimal symmetry $K$ whose associated Noether charge is linear in each tangent space. Our focus lies in investigating solutions to the Euler-Lagrange equations having fixed energy and that connect a given point $p$ to a flow line $\\gamma=\\gamma(t)$ of $K$ that does not cross $p$. By utilizing the invariance of $L$ under the flow of $K$, we simplify the problem into a two-point boundary problem. Consequently, we derive an equation that involves the differential of the ``arrival time'' $t$, seen as a functional on the infinite dimensional manifold of connecting paths satisfying the semi-holonomic constraint defined by the Noether charge. When $L$ is positively homogeneous of degree $2$ in the velocities, the resulting equation establishes a variational principle that extends the Fermat's principle in a stationary spacetime. Furthermore, we also analyze the scenario where the Noether charge is affine.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-15T21:01:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "affine noether charge",
      "fixed energy solutions",
      "euler-lagrange equations",
      "indefinite lagrangian",
      "infinite dimensional manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}