{
  "id": "2004.03526",
  "version": "v1",
  "published": "2020-04-07T16:32:21.000Z",
  "updated": "2020-04-07T16:32:21.000Z",
  "title": "Hamiltonian formalism of the inverse problem using Dirac geometry and its application on Linear systems",
  "authors": [
    "Hassan Najafi Alishah"
  ],
  "categories": [
    "math.DS",
    "math.SG"
  ],
  "abstract": "We present the Hamiltonian formalism for the inverse problem having Dirac$\\backslash$ big-isotropic structures as underlying geometry. We used the same idea at [1] to treat replicator equations. Here we state the procedure used there for general vector fields that can be written in a gradient form. For a linear system, we show that if representing matrix of the system has at least one pair of positive-negative non-zero eigenvalues or in the case of eigenvalue zero, at least one three dimensional Jordan block associated to it, then the linear system has a Hamiltonian description with respect to a non-trivial Dirac$\\backslash$big-isotropic structure. More interestingly, we prove that every Hamiltonian linear system is Hamiltonian integrable. As a byproduct, we found a class of linear systems with eigenvalue zero that are Hamiltonian only with respect to a proper big-isotropic structure. Our approach provides, also, a clear picture for the alternative Hamiltonian descriptions of linear systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-07T16:32:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "inverse problem",
      "hamiltonian formalism",
      "dirac geometry",
      "application",
      "hamiltonian description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}