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Multiplicity of periodic solution with prescribed energy to singular dynamical systems. (English) Zbl 0771.34035
Using the appropriate homotopical (pseudo-)index technique, multiplicity results are obtained on the existence of periodic solutions with prescribed energy for a class of dynamical systems with singular potentials. Both the cases, i.e. those of strong as well as of weak forces related to the given potential, are separately treated. The noncollision solutions, when the singularity is not crossed, are taken into account.
Reviewer: J.Andres (Olomouc)

34C25 Periodic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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[1] Ambrosetti, A.; Coti Zelati, V., Critical points with lack of compactness and applications to singular dynamical systems, Ann. di Matem. Pura e Appl., Vol. CIL, 237-259, (1987), Ser. IV · Zbl 0642.58017
[2] Ambrosetti, A.; Coti Zelati, V., Noncollision orbits for a class of Keplerian-like potentials, Ann. I.H.P. Analyse non linéaire, Vol. 5, 287-295, (1988) · Zbl 0667.58055
[3] Ambrosetti, A.; Coti Zelati, V., Perturbations of Hamiltonian systems with Keplerian potentials, Math. Zeit., Vol. 201, 227-242, (1989) · Zbl 0653.34032
[4] A. Ambrosetti and V. Coti Zelati, Closed Orbits of Fixed Energy for Singular Hamiltonian Systems, Preprint. · Zbl 0737.70008
[5] Bahri, A.; Rabinowitz, P. H., A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal., Vol. 82, 412-428, (1989) · Zbl 0681.70018
[6] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical points theory and application to some nonlinear problems with “strong” resonance at infinity, Nonlin. anal. T.M.A., Vol. 7, 981-1012, (1983) · Zbl 0522.58012
[7] Benci, V., A geometrical index for a group S^1 and some applications to the study of periodic solutions of O.D.E., Comm. Pure Appl. Math., Vol. 34, 393-432, (1981) · Zbl 0447.34040
[8] Benci, V., On the critical point theory for indefinite functionals in presence of symmetries, Trans. Am. Math. Soc., Vol. 274, 533-572, (1982) · Zbl 0504.58014
[9] Benci, V.; Giannoni, F., Periodic solutions of prescribed energy for a class of Hamiltonian systems with singular potentials, J. Differential equations, Vol. 82, 60-70, (1989) · Zbl 0689.34034
[10] A. Capozzi, S. Solimini and S. Terracini, On a Class of Dynamical Systems with Singular Potentials, Preprint S.I.S.S.A., Nonlin. Anal. T.M.A. (to appear). · Zbl 0744.34039
[11] Coti Zelati, V., Dynamical systems with effective-like potentials, Nonlin. Anal. T.M.A., Vol. 12, 209-222, (1988) · Zbl 0648.34050
[12] Coti Zelati, V., Periodic solutions for a class of planar, singular dynamical systems, J. Math. Pures et Appl., T. 68, 109-119, (1989) · Zbl 0633.34034
[13] Degiovanni, M.; Giannoni, F., Dynamical systems with Newtonian type potentials, Ann. Scuola Norm. Sup Pisa, Cl. Sci., Vol. 4, (1989), (to appear) · Zbl 0682.34031
[14] Degiovanni, M.; Giannoni, F.; Marino, A., Periodic solutions of dynamical systems with Newtonian type potentials, Atti Accad. Naz. Lincei, Rend. Cl. Sc. Fis. Mat. Nat., Vol. 81, 271-278, (1987) · Zbl 0667.70010
[15] Gordon, W., A minimizing property of Keplerian orbits, Amer. J. Math., Vol. 99, 961-971, (1975) · Zbl 0378.58006
[16] Gordon, W., Conservative dynamical systems involving strong forces, Trans. A.M.S., Vol. 204, 113-135, (1975) · Zbl 0276.58005
[17] Greco, C., Periodic solutions of a class of singular Hamiltonian systems, Nonlin. Anal. T.M.A., Vol. 12, 259-269, (1988) · Zbl 0648.34048
[18] Krasnoselski, M. A., On the estimation of the number of the critical points of functionals, Uspeki Math. Nank., Vol. 7, No. 2, 157-164, (1952), (48)
[19] L. Ljusternik and L. Schnirelmann, Méthodes topologiques dans les problèmes variaionnels, Hermann, Paris, 1934.
[20] Moser, J., Regularizaron of kepler’s problem and the averaging method on a maniold, Comm. Pure Appl. Math., Vol. 23, 609-636, (1970) · Zbl 0193.53803
[21] S. Terracini An Homotopical Index and Multiplicity of Periodic Solutions to Dynamical Systems with Singular Potentials, J. of Diff Eq. (to appear). · Zbl 0774.34028
[22] S. Terracini, Second Order Conservative Systems with Singular Potentials: Noncollision Periodic Solutions to the Fixed Energy Problem, Preprint, 1990.
[23] S. Terracini, Ph. D. Thesis, Preprint S.I.S.S.A., Trieste, 1990.
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