{
  "id": "1605.07864",
  "version": "v1",
  "published": "2016-05-25T12:59:04.000Z",
  "updated": "2016-05-25T12:59:04.000Z",
  "title": "Periodic solutions of singular first-order Hamiltonian systems of N-vortex type",
  "authors": [
    "Thomas Bartsch"
  ],
  "comment": "10 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We are concerned with the dynamics of $N$ point vortices $z_1,\\dots,z_N\\in\\Omega\\subset\\mathbb{R}^2$ in a planar domain. This is described by a Hamiltonian system \\[ \\Gamma_k\\dot{z}_k(t)=J\\nabla_{z_k} H\\big(z(t)\\big),\\quad k=1,\\dots,N, \\] where $\\Gamma_1,\\dots,\\Gamma_N\\in\\mathbb{R}\\setminus\\{0\\}$ are the vorticities, $J\\in\\mathbb{R}^{2\\times2}$ is the standard symplectic $2\\times2$ matrix, and the Hamiltonian $H$ is of $N$-vortex type: \\[ H(z_1,\\dots,z_N) = -\\frac1{2\\pi}\\sum_{j\\ne k}^N \\Gamma_j\\Gamma_k\\log|z_j-z_k| - \\sum_{j,k=1}^N\\Gamma_j\\Gamma_kg(z_j,z_k). \\] Here $g:\\Omega\\times\\Omega\\to\\mathbb{R}$ is an arbitrary symmetric function of class $C^2$, e.g.\\ the regular part of a hydrodynamic Green function. Given a nondegenerate critical point $a_0\\in\\Omega$ of $h(z)=g(z,z)$ and a nondegenerate relative equilibrium $Z(t)\\in\\mathbb{R}^{2N}$ of the Hamiltonian system in the plane with $g=0$, we prove the existence of a smooth path of periodic solutions $z^{(r)}(t)=\\big(z^{(r)}_1(t),\\dots,z^{(r)}_N(t)\\big)\\in\\Omega^N$, $0<r<r_0$, with $z^{(r)}_k(t)\\to a_0$ as $r\\to0$. In the limit $r\\to0$, and after a suitable rescaling, the solutions look like $Z(t)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-25T12:59:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37J45",
      "37N10",
      "76B47"
    ],
    "keywords": [
      "singular first-order hamiltonian systems",
      "periodic solutions",
      "n-vortex type",
      "arbitrary symmetric function",
      "hydrodynamic green function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}