Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds
Kazunaga Tanaka
Department of Mathematics, School of Science and Engineering, Waseda University 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169, Japan
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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:
under the conditions: (i) for all ; (ii) as and .
For closed geodesics, we show the existence of a non-constant closed geodesic on under the condition:
where is the standard metric on .
Résumé
Nous étudions l’existence de solutions p’eriodiques pour des systèmes Hamiltoniens singuliers, et de géodésiques fermées sur des variétes Riemanniennes non-compactes par des méthodes variationnelles.
Pour les systèmes Hamiltoniens, nouns montrons l’existence d’une solution périodique pour un probl‘eme à énergie prescrite:
sous les conditions: (i) pour tout ; (ii) quand et .
Pour les géodésiques fermées, nouns montrons l’existence d’une géodésique fermée non-constante sur sous la condition:
où est la métrique standard sur .
Cite this article
Kazunaga Tanaka, Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 1, pp. 1–33
DOI 10.1016/S0294-1449(99)00102-X