### 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 ∈ ℝ^{N} \ {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 × S^{N-1}, g) under the condition: g_{(s,x)} ∼ ds^{2} + h_{0} as s ∼ ± ∞, where h_{0} is the standard metric on S^{N-1}.

Original language | English |
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Pages (from-to) | 1-33 |

Number of pages | 33 |

Journal | Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis |

Volume | 17 |

Issue number | 1 |

Publication status | Published - 2000 Jan |

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### ASJC Scopus subject areas

- Analysis

### Cite this

**Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds.** / Tanaka, Kazunaga.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds

AU - Tanaka, Kazunaga

PY - 2000/1

Y1 - 2000/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0005827578&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0005827578&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0005827578

VL - 17

SP - 1

EP - 33

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 1

ER -