American Institute of Mathematical Sciences

Advanced
May  2017, 37(5): 2705-2715. doi: 10.3934/dcds.2017116

Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems

Mitsuru Shibayama ,

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 $α >1$.

Keywords: Periodic solutions, Hamiltonian systems, variational problems.
Mathematics Subject Classification: Primary: 70H12, 70F05; Secondary: 34C25.
Citation: Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete & Continuous Dynamical Systems, 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

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

Download full-size image

Download as PowerPoint slide

[1]

Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069
[2]

Alessandro Fonda, Andrea Sfecci. Multiple periodic solutions of Hamiltonian systems confined in a box. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1425-1436. doi: 10.3934/dcds.2017059
[3]

Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835
[4]

Jianshe Yu, Honghua Bin, Zhiming Guo. Periodic solutions for discrete convex Hamiltonian systems via Clarke duality. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 939-950. doi: 10.3934/dcds.2006.15.939
[5]

Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017
[6]

Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361
[7]

Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 251-264. doi: 10.3934/dcds.1997.3.251
[8]

Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222
[9]

Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024
[10]

Giuseppe Cordaro. Existence and location of periodic solutions to convex and non coercive Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 983-996. doi: 10.3934/dcds.2005.12.983
[11]

Juhong Kuang, Weiyi Chen, Zhiming Guo. Periodic solutions with prescribed minimal period for second order even Hamiltonian systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021166
[12]

Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9
[13]

V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277
[14]

Xiao-Fei Zhang, Fei Guo. Multiplicity of subharmonic solutions and periodic solutions of a particular type of super-quadratic Hamiltonian systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1625-1642. doi: 10.3934/cpaa.2016005
[15]

Chungen Liu, Xiaofei Zhang. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1559-1574. doi: 10.3934/dcds.2017064
[16]

Xingyong Zhang, Xianhua Tang. Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems. Communications on Pure & Applied Analysis, 2014, 13 (1) : 75-95. doi: 10.3934/cpaa.2014.13.75
[17]

Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268
[18]

Jean Mawhin. Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 4015-4026. doi: 10.3934/dcds.2012.32.4015
[19]

Weigao Ge, Li Zhang. Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4925-4943. doi: 10.3934/dcds.2016013
[20]

S. Aubry, G. Kopidakis, V. Kadelburg. Variational proof for hard Discrete breathers in some classes of Hamiltonian dynamical systems. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 271-298. doi: 10.3934/dcdsb.2001.1.271

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (76)
  • HTML views (66)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences