Study of the periodic solutions of some \(n\)-body-type problems. (English) Zbl 0987.70011
The author considers possible periodic solutions of the equations of motion of an \(n\)-body system under no external forces and with periodic internal forces. He considers the possibility of collisions, and points out the importance of Morse index of critical points of Hamilton’s principal function. He also discusses in detail the fixed period case and the fixed energy case.
Reviewer: Ll.G.Chambers (Bangor)
Keywords:
collision; \(n\)-body problem; Hamilton’s function; periodic solutions; periodic internal forces; Morse index; critical points; fixed period case; fixed energy caseReferences:
| [1] | Ambrosetti, A.; Coti-Zelati, V., Non collision periodic solution for a class of symmetric 3-body type problems, Topological Method. Nonlinear Anal., 3, 197-207 (1994) · Zbl 0829.70006 |
| [2] | A. Ambrosetti, V. Coti-Zelati, Periodic Solutions of Singular Lagrangian Systems, Birkhaüser, Basel, 1994.; A. Ambrosetti, V. Coti-Zelati, Periodic Solutions of Singular Lagrangian Systems, Birkhaüser, Basel, 1994. · Zbl 0829.70006 |
| [3] | Ambrosetti, A.; Coti-Zelati, V., Non-collision orbits for a class of Keplerian-like potentials, Ann. Inst. Henri Poincaré, 5, 3, 287-295 (1988) · Zbl 0667.58055 |
| [4] | Ambrosetti, A.; Struwe, M., Periodic motions of conservative systems with singular potentials, Nonlinear Differential equations Appl., 1, 179-202 (1994) · Zbl 0821.34036 |
| [5] | Ambrosetti, A.; Tanaka, K.; Vitillaro, E., Periodic solution with prescribed energy for some Keplerian n-body problems, Ann. Inst. Henri Poincaré, 11, 6, 614-632 (1994) · Zbl 0855.70006 |
| [6] | Bahri, A.; D’Onofrio, B., Exponential growth of the number of periodic orbits for three-body type problems, Maghreb Math. Rev., 1, 1, 1-14 (1992) · Zbl 0801.70006 |
| [7] | Bahri, A.; Lions, P. L., Morse index of some min-max critical points I. Applications to multiplicity results, Comm. Pure Appl. Math, 41, 1027-1037 (1988) · Zbl 0645.58013 |
| [8] | Bahri, A.; Rabinowitz, P. H., A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal., 82, 412-428 (1989) · Zbl 0681.70018 |
| [9] | Bahri, A.; Rabinowitz, A., Periodic solutions of Hamiltonian systems of 3-body type, Ann. IHP, Anal. Non linaire, 8, 561-649 (1991) · Zbl 0745.34034 |
| [10] | Beaulieu, A., Etude des solutions generalisees pour un systeme Hamiltonien avec potentiel singulier, Duke Math. J., 67, 21-37 (1992) · Zbl 0758.35016 |
| [11] | Coti Zelati, V., Periodic solutions for a class of planar singular dynamical systems, J. Math. Pure Appl., 68, 109-119 (1989) · Zbl 0633.34034 |
| [12] | Coti Zelati, V., Periodic solutions for N-body problems, Ann. Inst. Henri Poincaré, 7, 5, 477-492 (1990) · Zbl 0723.70010 |
| [13] | Coti Zelati, V.; Serra, E., Collisions and non collisions solutions for a class of Keplerian-like dynamical systems, Ann. Mat. Pura. Appl., 166, 4, 343-362 (1994) · Zbl 0832.70009 |
| [14] | Degiovanni, M.; Giannoni, F., Dynamical systems with Newtonian type potentials, Ann. Scuola Norm. Sup. Pisa cl. Sci., 4, 15, 267-294 (1988) |
| [15] | G. Dell’Antonio, Classical solutions for a perturbed N-body system, in: A. Matzeu (Ed.), Topological Nonlinear Analysis II, Birkhaüser, Vignolied, 1997, pp. 1-86.; G. Dell’Antonio, Classical solutions for a perturbed N-body system, in: A. Matzeu (Ed.), Topological Nonlinear Analysis II, Birkhaüser, Vignolied, 1997, pp. 1-86. · Zbl 0895.70009 |
| [16] | N. Ghoussoub, Duality perturbation method in critical point theory, Cambridge Tracts in Mathematics, Cambridge University Press, 1993.; N. Ghoussoub, Duality perturbation method in critical point theory, Cambridge Tracts in Mathematics, Cambridge University Press, 1993. · Zbl 0790.58002 |
| [17] | Lazer, A. C.; Solimini, S., Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Nonlinear Anal. TMA, 12, 761-775 (1988) · Zbl 0667.47036 |
| [18] | Majer, P., Variational methods on manifolds with boundary, Topology, 34, 1, 1-12 (1995) · Zbl 0819.58003 |
| [19] | P. Majer, S. Terracini, Multiple periodic solutions to some n-body type problems via collision index, preprint, 1993.; P. Majer, S. Terracini, Multiple periodic solutions to some n-body type problems via collision index, preprint, 1993. · Zbl 0782.70010 |
| [20] | Majer, P.; Terracini, S., Periodic solutions to some problems of n-body type, Arch. Rational. Mech. Anal., 1124, 381-404 (1993) · Zbl 0782.70010 |
| [21] | Majer, P.; Terracini, S., Periodic solutions to some problems of n-body type, the fixed energy case, Duke Math. J., 69, 3, 683-697 (1993) · Zbl 0807.70009 |
| [22] | Marino, A.; Prodi, G., Metodi perturbativi nella teoria di Morse, Boll. Un. ITAL, 11, 1-32 (1975) · Zbl 0311.58006 |
| [23] | Ramos, M.; Terracini, S., Noncollision periodic solutions to some singular systems with very weak forces, J. Differential Equations, 118, 1, 121-152 (1995) · Zbl 0826.34039 |
| [24] | Riahi, H., Periodic orbits of n body type problems: the fixed period case, Trans. Amer. Math. Soc., 347, 1, 4663-4685 (1995) · Zbl 0849.58057 |
| [25] | Riahi, H.; of the generalized solutions of n-body type problems with weak force, Study, Nonlinear Anal. Theory, Methods Appl., 28, 1, 49-59 (1997) · Zbl 0914.70007 |
| [26] | H. Riahi, Study of the critical points at infinity arising from the failure of the Palais-Smale condition for n-body type problems, Mem. Amer. Math. Soc. 1999.; H. Riahi, Study of the critical points at infinity arising from the failure of the Palais-Smale condition for n-body type problems, Mem. Amer. Math. Soc. 1999. · Zbl 0928.58018 |
| [27] | H. Riahi, Exponential growth of the number of periodic orbits for N-body type problems with fixed energy, in preparation.; H. Riahi, Exponential growth of the number of periodic orbits for N-body type problems with fixed energy, in preparation. |
| [28] | Serra, E.; Terracini, S., Collisionless periodic solutions to some 3-body problems, Arch. Rational Mech. Anal., 120, 305-325 (1992) · Zbl 0773.70009 |
| [29] | Serra, E.; Terracini, S., Non-collision periodic solutions to some singular minimization problems with Keplerian like potentials, Nonlinear Anal. Theory Methods Appl., 22, 45-62 (1994) · Zbl 0813.70006 |
| [30] | Tanaka, K., Morse indices at critical points related to the symmetric Mountain-Pas Theorem and application, Comm. Partial Differential Equations, 14, 1, 99-128 (1989) · Zbl 0669.34035 |
| [31] | Tanaka, K., Noncollision solutions for a second order singular Hamiltonian system with weak force, Ann. Inst. Henri Poincaré, Anal. Nonlinéaire, 10, 2, 215-238 (1993) · Zbl 0781.58036 |
| [32] | Tanaka, K., A prescribed energy problem for a singular Hamiltonian system with a weak force, J. Funct. Anal., 113, 2, 351-390 (1993) · Zbl 0771.70014 |
| [33] | Tanaka, K., A prescribed energy problem for a conservative singular Hamiltonian system, Arch. Rational Mech. Anal., 128, 127-164 (1994) · Zbl 0823.34047 |
| [34] | Viterbo, C., Indice de Morse des points critiques obtenus par minimax, Ann. Inst. Henri Poincaré, 5, 3, 221-225 (1988) · Zbl 0695.58007 |
| [35] | E. Vitillaro, Noncollision periodic solution of fixed energy for a symmetric N-body type problems, in: Variational and Local Methods in the Study of Hamiltonian Systems (Trieste, 1994), World Sci. Publishing, River Edge, NJ, 1995, pp. 202-211.; E. Vitillaro, Noncollision periodic solution of fixed energy for a symmetric N-body type problems, in: Variational and Local Methods in the Study of Hamiltonian Systems (Trieste, 1994), World Sci. Publishing, River Edge, NJ, 1995, pp. 202-211. · Zbl 0963.70545 |
| [36] | S.Q. Zhang, Q. Zhou, Symmetric periodic non collision solutions for N-body problems (English summary), Acta. Math. Sinica 39 (2) (1996), [Acta. Math. Sinica (N.S) 11 (1) (1995), 37-43 (in Chinese)].; S.Q. Zhang, Q. Zhou, Symmetric periodic non collision solutions for N-body problems (English summary), Acta. Math. Sinica 39 (2) (1996), [Acta. Math. Sinica (N.S) 11 (1) (1995), 37-43 (in Chinese)]. · Zbl 0836.70008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
