{
  "id": "2304.10205",
  "version": "v1",
  "published": "2023-04-20T10:48:10.000Z",
  "updated": "2023-04-20T10:48:10.000Z",
  "title": "A Modified Parameterization Method for Invariant Lagrangian Tori for Partially Integrable Hamiltonian Systems",
  "authors": [
    "Jordi-Lluís Figueras",
    "Alex Haro"
  ],
  "comment": "39 pages",
  "categories": [
    "math.DS",
    "nlin.CD"
  ],
  "abstract": "In this paper we present an a-posteriori KAM theorem for the existence of an $(n-d)$-parameters family of $d$-dimensional isotropic invariant tori with Diophantine frequency vector $\\omega\\in \\mathbb R^d$, of type $(\\gamma,\\tau)$, for $n$ degrees of freedom Hamiltonian systems with $(n-d)$ independent first integrals in involution. If the first integrals induce a Hamiltonian action of the $(n-d)$-dimensional torus, then we can produce $n$-dimensional Lagrangian tori with frequency vector of the form $(\\omega,\\omega_p)$, with $\\omega_p\\in\\mathbb R^{n-d}$. In the light of the parameterization method, we design a (modified) quasi-Newton method for the invariance equation of the parameterization of the torus, whose proof of convergence from an initial approximation, and under appropriate non-degeneracy conditions, is the object of this paper. We present the results in the analytic category, so the initial torus is real-analytic in a certain complex strip of size $\\rho$, and the corresponding error in the functional equation is $\\varepsilon$. We heavily use geometric properties and the so called automatic reducibility to deal directly with the functional equation and get convergence if $\\gamma^{-2} \\rho^{-2\\tau-1}\\varepsilon$ is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if $\\gamma^{-4} \\rho^{-4\\tau}\\varepsilon$ is small enough. The approach is suitable to perform computer assisted proofs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-20T10:48:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partially integrable hamiltonian systems",
      "invariant lagrangian tori",
      "modified parameterization method",
      "functional equation",
      "first integrals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}