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Homework Help: Periodic orbit of Hamiltonian dynamical system in R^2 is stable

  1. Jun 11, 2010 #1
    1. The problem statement, all variables and given/known data
    We are given a Hamiltonian dynamical system with a smooth Hamiltonian [tex]H:\mathbb{R}^2\to\mathbb{R}[/tex] on [tex]\mathbb{R}^2[/tex], with canonical symplectic structure. Suppose this Hamiltonian has a periodic orbit [tex]H^{-1}(h_0)[/tex]. Prove that there exists an [tex]\epsilon>0[/tex] such that for all [tex]h\in ]h_0-\epsilon,h_0+\epsilon[[/tex]: [tex]H^{-1}(h)[/tex] is also a periodic orbit.

    2. Relevant equations
    Liouville's theorem tells us that the flow of a Hamiltonian vector field preserves the Liouville volume form in [tex]\mathbb{R}^2[/tex].

    3. The attempt at a solution
    It is enough to prove that [tex]H^{-1}(h)[/tex] is bounded. It then follows from Liouville's theorem that it is also a periodic orbit. However, I have no idea how to do this.

    Thanks in advance!
  2. jcsd
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