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

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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.

Original languageEnglish
Pages (from-to)1-33
Number of pages33
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume17
Issue number1
DOIs
Publication statusPublished - 2000 Jan

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ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

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