# Non-collision solutions for a second order singular hamiltonian system with weak force

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Non-collision solutions for a second order singular Hamiltonian system with weak force ANNALES DE L’I. H. P., SECTION C KAZUNAGA TANAKA Non-collision solutions for a second order singular Hamiltonian system with weak force Annales de l’I. H. P., section C, tome 10, no 2 (1993), p. 215-238. <http://www.numdam.org/item?id=AIHPC_1993__10_2_215_0> © Gauthier-Villars, 1993, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section C » (http://www.elsevier.com/locate/anihpc), implique l’accord avec les condi- tions générales d’utilisation (http://www.numdam.org/legal.php). Toute uti- lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Non-collision solutions for a second order singular Hamiltonian system with weak force Kazunaga TANAKA Department of Mathematics, School of Science, Nagoya University, Chikusa-ku, Nagoya 464, Japan Ann. Inst. Henri Poincaré, Vol. 10, n° 2, 1993, p. 215-238. Analyse non linéaire ABSTRACT. - Under a weak force type condition, we consider the existence of time periodic solutions of singular Hamiltonian systems: We assume V(q, t)O for all q, t and V {q, t), Vq(q, t) -+ 0 as Moreover we assume V (q, t) is of a form: where 0 oc ~ ~ and L~ (q, C2 ((RNB~ 0 ~~ ~ R, R) is aT-periodic func- tion in t such that For oc E ( l, 2], we prove the existence of a non-collision solution of (HS). For 1], we prove that the generalized solution of(HS), which is introduced in [BR], enters the singularity 0 at most one time in its period. Our argument depends on a minimax argument due to [BR] and an estimate of Morse index of corresponding functional, which will be obtai- ned via re-scaling argument. Key words : Periodic solutions, Hamiltonian systems, singular potentials, minimax metho

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