{
  "id": "1701.08026",
  "version": "v1",
  "published": "2017-01-27T12:21:56.000Z",
  "updated": "2017-01-27T12:21:56.000Z",
  "title": "Curvature in Hamiltonian Mechanics And The Einstein-Maxwell-Dilaton Action",
  "authors": [
    "S. G. Rajeev"
  ],
  "comment": "2 figs",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP"
  ],
  "abstract": "Riemannian geometry is a particular case of Hamiltonian mechanics: the orbits of the hamiltonian $H=\\frac{1}{2}g^{ij}p_{i}p_{j}$ are the geodesics. Given a symplectic manifold (\\Gamma,\\omega), a hamiltonian $H:\\Gamma\\to\\mathbb{R}$ and a Lagrangian sub-manifold $M\\subset\\Gamma$ we find a generalization of the notion of curvature. The particular case $H=\\frac{1}{2}g^{ij}\\left[p_{i}-A_{i}\\right]\\left[p_{j}-A_{j}\\right]+\\phi $ of a particle moving in a gravitational, electromagnetic and scalar fields is studied in more detail. The integral of the generalized Ricci tensor w.r.t. the Boltzmann weight reduces to the action principle $\\int\\left[R+\\frac{1}{4}F_{ik}F_{jl}g^{kl}g^{ij}-g^{ij}\\partial_{i}\\phi\\partial_{j}\\phi\\right]e^{-\\phi}\\sqrt{g}d^{n}q$ for the scalar, vector and tensor fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-27T12:21:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hamiltonian mechanics",
      "einstein-maxwell-dilaton action",
      "boltzmann weight reduces",
      "riemannian geometry",
      "tensor fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}