{
  "id": "2405.07410",
  "version": "v1",
  "published": "2024-05-13T01:09:40.000Z",
  "updated": "2024-05-13T01:09:40.000Z",
  "title": "Non-unique Hamiltonians for Discrete Symplectic Dynamics",
  "authors": [
    "Liyan Ni",
    "Yihao Zhao",
    "Zhonghan Hu"
  ],
  "comment": "4 pages, 1 figure",
  "categories": [
    "math-ph",
    "math.MP",
    "physics.chem-ph",
    "physics.comp-ph"
  ],
  "abstract": "An outstanding property of any Hamiltonian system is the symplecticity of its flow, namely, the continuous trajectory preserves volume in phase space. Given a symplectic but discrete trajectory generated by a transition matrix applied at a fixed time-increment ($\\tau > 0$), it was generally believed that there exists a unique Hamiltonian producing a continuous trajectory that coincides at all discrete times ($t = n\\tau$ with $n$ integers) as long as $\\tau$ is small enough. However, it is now exactly demonstrated that, for any given discrete symplectic dynamics of a harmonic oscillator, there exists an infinite number of real-valued Hamiltonians for any small value of $\\tau$ and an infinite number of complex-valued Hamiltonians for any large value of $\\tau$. In addition, when the transition matrix is similar to a Jordan normal form with the supradiagonal element of $1$ and the two identical diagonal elements of either $1$ or $-1$, only one solution to the Hamiltonian is found for the case with the diagonal elements of $1$, but no solution can be found for the other case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-13T01:09:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "discrete symplectic dynamics",
      "non-unique hamiltonians",
      "infinite number",
      "transition matrix",
      "jordan normal form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}