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May
2017, 37(5): 2705-2715.
doi: 10.3934/dcds.2017116
Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems
Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku Kyoto 606-8501, Japan |
We study the existence of periodic solutions for a prescribed-energy problem of Hamiltonian systems whose potential function has a singularity at the origin like $-1/|q|^{α} (q ∈ \mathbb{R}^N)$. It is known that there exist generalized periodic solutions which may have collisions, and the number of possible collisions has been estimated. In this paper we obtain a new estimation of the number of collisions. Especially we show that the obtained solutions have no collision if $N ≥ 2$ and $α >1$.
Mathematics Subject Classification:
Primary: 70H12, 70F05; Secondary: 34C25.
Citation:
Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete and Continuous Dynamical Systems,
2017, 37
(5)
: 2705-2715.
doi: 10.3934/dcds.2017116
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References:
| [1] |
A. Ambrosetti and V. Coti-Zelati,
Periodic Solutions of Singular Lagrangian Systems,
Birkhauser, 1993.
doi: 10.1007/978-1-4612-0319-3. |
| [2] |
A. Ambrosetti and V. Coti-Zelati,
Closed orbits of fixed energy for singular Hamiltonian systems, Arch. Rational Mech. Anal., 112 (1990), 339-362.
doi: 10.1007/BF02384078. |
| [3] |
H. Hofer and E. Zehnder,
Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math., 90 (1987), 1-9.
doi: 10.1007/BF01389030. |
| [4] |
T. J. Hunt and R. S. MacKay,
Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor, Nonlinearity, 16 (2003), 1499-1510.
doi: 10.1088/0951-7715/16/4/318. |
| [5] |
J. Milnor,
On the geometry of the Kepler problem, Amer. Math. Monthly, 90 (1983), 353-365.
doi: 10.2307/2975570. |
| [6] |
R. Montgomery,
Fitting hyperbolic pants to a three-body problem, Ergodic Theory Dynam. Systems, 25 (2005), 921-947.
doi: 10.1017/S0143385704000653. |
| [7] |
P. H. Rabinowitz,
Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184.
doi: 10.1002/cpa.3160310203. |
| [8] |
K. Tanaka,
A prescribed energy problem for a singular Hamiltonian system with a weak force, J. Funct. Anal., 113 (1993), 351-390.
doi: 10.1006/jfan.1993.1054. |
| [9] |
C. Viterbo,
A proof of Weinstein's conjecture in $\mathbb{R}^{2n}$, Annales de l'institut Henri Poincaré (C) Analyse non linéaire, 4 (1987), 337-356.
|
| [10] |
A. Weinstein,
Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2), 108 (1978), 507-518.
doi: 10.2307/1971185. |
show all references
References:
| [1] |
A. Ambrosetti and V. Coti-Zelati,
Periodic Solutions of Singular Lagrangian Systems,
Birkhauser, 1993.
doi: 10.1007/978-1-4612-0319-3. |
| [2] |
A. Ambrosetti and V. Coti-Zelati,
Closed orbits of fixed energy for singular Hamiltonian systems, Arch. Rational Mech. Anal., 112 (1990), 339-362.
doi: 10.1007/BF02384078. |
| [3] |
H. Hofer and E. Zehnder,
Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math., 90 (1987), 1-9.
doi: 10.1007/BF01389030. |
| [4] |
T. J. Hunt and R. S. MacKay,
Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor, Nonlinearity, 16 (2003), 1499-1510.
doi: 10.1088/0951-7715/16/4/318. |
| [5] |
J. Milnor,
On the geometry of the Kepler problem, Amer. Math. Monthly, 90 (1983), 353-365.
doi: 10.2307/2975570. |
| [6] |
R. Montgomery,
Fitting hyperbolic pants to a three-body problem, Ergodic Theory Dynam. Systems, 25 (2005), 921-947.
doi: 10.1017/S0143385704000653. |
| [7] |
P. H. Rabinowitz,
Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184.
doi: 10.1002/cpa.3160310203. |
| [8] |
K. Tanaka,
A prescribed energy problem for a singular Hamiltonian system with a weak force, J. Funct. Anal., 113 (1993), 351-390.
doi: 10.1006/jfan.1993.1054. |
| [9] |
C. Viterbo,
A proof of Weinstein's conjecture in $\mathbb{R}^{2n}$, Annales de l'institut Henri Poincaré (C) Analyse non linéaire, 4 (1987), 337-356.
|
| [10] |
A. Weinstein,
Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2), 108 (1978), 507-518.
doi: 10.2307/1971185. |
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