# American Institute of Mathematical Sciences

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 [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

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
Graph of $f(1, \alpha, \alpha, \alpha)$.
 [1] Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 [2] Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030 [3] Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170 [4] João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 [5] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [6] Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033 [7] Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379 [8] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [9] Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215 [10] Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469 [11] Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240 [12] Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 [13] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [14] Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385 [15] Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012 [16] Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446 [17] Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434 [18] Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451 [19] Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029 [20] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

2019 Impact Factor: 1.338