Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds

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    15 Citations (Scopus)

    Abstract

    We study the existence of periodic solutions of singular Hamiltonian systems as well as closed geodesics on non-compact Riemannian manifolds via variational methods. For Hamiltonian systems, we show the existence of a periodic solution of prescribed-energy problem: q̇̇+∇V(q)=0, 1/2|q̇|2+V(q)=0 under the conditions: (i) V(q) < 0 for all q ∈ ℝN \ {0}; (ii) V(q) ∼ -1/|q|2 as |q| ∼ 0 and \q\ ~∼ ∞. For closed geodesics, we show the existence of a non-constant closed geodesic on (ℝ × SN-1, g) under the condition: g(s,x) ∼ ds2 + h0 as s ∼ ± ∞, where h0 is the standard metric on SN-1.

    Original languageEnglish
    Pages (from-to)1-33
    Number of pages33
    JournalAnnales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis
    Volume17
    Issue number1
    Publication statusPublished - 2000 Jan

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    Hamiltonians
    Closed Geodesics
    Noncompact Manifold
    Singular Systems
    Hamiltonian Systems
    Riemannian Manifold
    Periodic Solution
    Variational Methods
    Metric
    Energy

    ASJC Scopus subject areas

    • Analysis

    Cite this

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    abstract = "We study the existence of periodic solutions of singular Hamiltonian systems as well as closed geodesics on non-compact Riemannian manifolds via variational methods. For Hamiltonian systems, we show the existence of a periodic solution of prescribed-energy problem: q̇̇+∇V(q)=0, 1/2|q̇|2+V(q)=0 under the conditions: (i) V(q) < 0 for all q ∈ ℝN \ {0}; (ii) V(q) ∼ -1/|q|2 as |q| ∼ 0 and \q\ ~∼ ∞. For closed geodesics, we show the existence of a non-constant closed geodesic on (ℝ × SN-1, g) under the condition: g(s,x) ∼ ds2 + h0 as s ∼ ± ∞, where h0 is the standard metric on SN-1.",
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