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

* Corresponding author

Received  April 2016 Revised  December 2016 Published  February 2017

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 $α >1$.

Citation: Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2705-2715. doi: 10.3934/dcds.2017116
References:
[1]

A. Ambrosetti and V. Coti-Zelati, Periodic Solutions of Singular Lagrangian Systems, Birkhauser, 1993. doi: 10.1007/978-1-4612-0319-3. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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J. Milnor, On the geometry of the Kepler problem, Amer. Math. Monthly, 90 (1983), 353-365. doi: 10.2307/2975570. Google Scholar

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R. Montgomery, Fitting hyperbolic pants to a three-body problem, Ergodic Theory Dynam. Systems, 25 (2005), 921-947. doi: 10.1017/S0143385704000653. Google Scholar

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P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203. Google Scholar

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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. Google Scholar

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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. Google Scholar

[10]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2), 108 (1978), 507-518. doi: 10.2307/1971185. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

J. Milnor, On the geometry of the Kepler problem, Amer. Math. Monthly, 90 (1983), 353-365. doi: 10.2307/2975570. Google Scholar

[6]

R. Montgomery, Fitting hyperbolic pants to a three-body problem, Ergodic Theory Dynam. Systems, 25 (2005), 921-947. doi: 10.1017/S0143385704000653. Google Scholar

[7]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2), 108 (1978), 507-518. doi: 10.2307/1971185. Google Scholar

Figure 1.  Graph of $f(1, \alpha, \alpha, \alpha)$.
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