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April  2019, 24(4): 1617-1625. doi: 10.3934/dcdsb.2018222

Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems

Liang Ding 1, , Rongrong Tian 2, and  Jinlong Wei 3,,

1. 

School of Mathematics, Sichuan University, Chengdu 610064, China
2. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
3. 

School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China

* Corresponding author: Jinlong Wei

Received  January 2018 Revised  March 2018 Published  April 2019 Early access  June 2018

Fund Project: The first author is partially supported by National Science Foundation of China (116712787) and Graduate Student's Research and Innovation Fund of Sichuan University (2018YJSY045). The third author is partially supported by National Science Foundation of China (11501577, 61773401).

In the past years, there were very few works on the existence of nonconstant periodic solutions with fixed energy of singular second-order Hamiltonian systems, and now we attempt to ingeniously use Ekeland's variational principle to prove the existence of nonconstant periodic solutions with any fixed energy for singular second-order Hamiltonian systems, and our results greatly generalize some well known results such as [1,Theorem 3.6]. Moreover, we exhibit two simple and instructive singular potential examples to make our result more clear, which have not been solved by known results.

Keywords: Nonconstant periodic solutions, any fixed energy, singular Hamiltonian systems.
Mathematics Subject Classification: Primary: 70F15, 70H35; Secondary: 34C25, 58E05.
Citation: Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222
References:
[1]

A. Ambrosetti and V. C. Zelati, Closed orbits of fixed energy for singular Hamiltonian systems, Arch. Ration. Mech. Anal., 112 (1990), 339-362.  doi: 10.1007/BF02384078.

[2]

A. Ambrosetti and V. C. Zelati, Closed orbits of fixed energy for a class of N-body problems, Ann. Institute H. Poincaré-Anal. Non Linéaive, 9 (1992), 187-200.  doi: 10.1016/S0294-1449(16)30244-X.

[3]

A. Ambrosetti and V. C. Zelati, Periodic Solutions of Singular Lagrangian Systems, Birkäuser, Boston, 1993. doi: 10.1007/978-1-4612-0319-3.

[4]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 1997.

[5]

V. Benci, Normal modes of a Lagrangian system constrained in a potential well, Ann. Institute H. Poincaré-Anal. Non Linéaive, 1 (1984), 379-400.  doi: 10.1016/S0294-1449(16)30419-X.

[6]

V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273.  doi: 10.1007/BF01389883.

[7]

M. Boughariou, Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces, Discrete Contin. Dynam. Systems, 9 (2003), 603-616.  doi: 10.3934/dcds.2003.9.603.

[8]

C. Carminati, E. Sere and K. Tanaka, The fixed energy problem for a class of nonconvex singular Hamiltonian systems, J. Differential Equations, 230 (2006), 362-377.  doi: 10.1016/j.jde.2006.01.021.

[9]

R. Castelli, Topologically distinct collision-free periodic solutions for the $N$ -center problem, Arch. Ration. Mech. Anal., 223 (2017), 941-975.  doi: 10.1007/s00205-016-1049-0.

[10]

C. F. Che and X. P. Xue, Periodic solutions for second order Hamiltonian systems on an arbitrary energy surface, Ann. Pol. Math., 105 (2012), 1-12.  doi: 10.4064/ap105-1-1.

[11]

W. B. Gordon, Conservative dynamical systems involving strong forces, Trans. Am. Math. Soc., 204 (1975), 113-135.  doi: 10.1090/S0002-9947-1975-0377983-1.

[12]

Y. M. Long and S. Q. Zhang, Geometric characterization for variational minimization solutions of the 3-body problem with fixed energy, J. Differential Equations, 160 (2000), 422-438.  doi: 10.1006/jdeq.1999.3659.

[13]

P. Majer and S. Terracini, Periodic solutions to some N-body type problems: the fixed energy case, Duke Math. J., 69 (1993), 683-697.  doi: 10.1215/S0012-7094-93-06929-3.

[14]

P. Majer and S. Terracini, Periodic solutions to some problems of n-body type, Arch. Ration. Mech. Anal., 124 (1993), 381-404.  doi: 10.1007/BF00375608.

[15]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York-Berlin, 1989. doi: 10.1007/978-1-4757-2061-7.

[16]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-9323-5.

[17]

R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.

[18]

L. Pisani, Periodic solutions with prescribed energy for singular conservative systems involving strong forces, Nonlinear Anal. Theor., 21 (1993), 167-179.  doi: 10.1016/0362-546X(93)90107-4.

[19]

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

[20]

E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, Princeton, 2005.

[21]

S. Terracini, Multiplicity of periodic solution with prescribed energy to singular dynamical systems, Ann. Institute H. Poincaré-Anal. Non Linéaive, 9 (1992), 597-641.  doi: 10.1016/S0294-1449(16)30224-4.

[22]

E. Vitillaro, Periodic solutions for singular conservative systems, J. Math. Anal. Appl., 185 (1994), 403-429.  doi: 10.1006/jmaa.1994.1258.

[23]

D. L. Wu, C. Li and P. F. Yuan, Periodic solutions for a class of second-order Hamiltonian systems of precribed energy, Electron. J. Qual. Theo., 77 (2015), 1-10. 

[24]

S. Q. Zhang, Multiple closed orbits of fixed energy for N-body-type problems with gravitational potentials, J. Math. Anal. Appl., 208 (1997), 462-475.  doi: 10.1006/jmaa.1997.5338.

[25]

S. Q. Zhang, Periodic solutions for some second order Hamiltonian systems, Nonlinearity, 22 (2009), 2141-2150.  doi: 10.1088/0951-7715/22/9/005.

show all references

References:
[1]

A. Ambrosetti and V. C. Zelati, Closed orbits of fixed energy for singular Hamiltonian systems, Arch. Ration. Mech. Anal., 112 (1990), 339-362.  doi: 10.1007/BF02384078.

[2]

A. Ambrosetti and V. C. Zelati, Closed orbits of fixed energy for a class of N-body problems, Ann. Institute H. Poincaré-Anal. Non Linéaive, 9 (1992), 187-200.  doi: 10.1016/S0294-1449(16)30244-X.

[3]

A. Ambrosetti and V. C. Zelati, Periodic Solutions of Singular Lagrangian Systems, Birkäuser, Boston, 1993. doi: 10.1007/978-1-4612-0319-3.

[4]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 1997.

[5]

V. Benci, Normal modes of a Lagrangian system constrained in a potential well, Ann. Institute H. Poincaré-Anal. Non Linéaive, 1 (1984), 379-400.  doi: 10.1016/S0294-1449(16)30419-X.

[6]

V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273.  doi: 10.1007/BF01389883.

[7]

M. Boughariou, Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces, Discrete Contin. Dynam. Systems, 9 (2003), 603-616.  doi: 10.3934/dcds.2003.9.603.

[8]

C. Carminati, E. Sere and K. Tanaka, The fixed energy problem for a class of nonconvex singular Hamiltonian systems, J. Differential Equations, 230 (2006), 362-377.  doi: 10.1016/j.jde.2006.01.021.

[9]

R. Castelli, Topologically distinct collision-free periodic solutions for the $N$ -center problem, Arch. Ration. Mech. Anal., 223 (2017), 941-975.  doi: 10.1007/s00205-016-1049-0.

[10]

C. F. Che and X. P. Xue, Periodic solutions for second order Hamiltonian systems on an arbitrary energy surface, Ann. Pol. Math., 105 (2012), 1-12.  doi: 10.4064/ap105-1-1.

[11]

W. B. Gordon, Conservative dynamical systems involving strong forces, Trans. Am. Math. Soc., 204 (1975), 113-135.  doi: 10.1090/S0002-9947-1975-0377983-1.

[12]

Y. M. Long and S. Q. Zhang, Geometric characterization for variational minimization solutions of the 3-body problem with fixed energy, J. Differential Equations, 160 (2000), 422-438.  doi: 10.1006/jdeq.1999.3659.

[13]

P. Majer and S. Terracini, Periodic solutions to some N-body type problems: the fixed energy case, Duke Math. J., 69 (1993), 683-697.  doi: 10.1215/S0012-7094-93-06929-3.

[14]

P. Majer and S. Terracini, Periodic solutions to some problems of n-body type, Arch. Ration. Mech. Anal., 124 (1993), 381-404.  doi: 10.1007/BF00375608.

[15]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York-Berlin, 1989. doi: 10.1007/978-1-4757-2061-7.

[16]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-9323-5.

[17]

R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.

[18]

L. Pisani, Periodic solutions with prescribed energy for singular conservative systems involving strong forces, Nonlinear Anal. Theor., 21 (1993), 167-179.  doi: 10.1016/0362-546X(93)90107-4.

[19]

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

[20]

E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, Princeton, 2005.

[21]

S. Terracini, Multiplicity of periodic solution with prescribed energy to singular dynamical systems, Ann. Institute H. Poincaré-Anal. Non Linéaive, 9 (1992), 597-641.  doi: 10.1016/S0294-1449(16)30224-4.

[22]

E. Vitillaro, Periodic solutions for singular conservative systems, J. Math. Anal. Appl., 185 (1994), 403-429.  doi: 10.1006/jmaa.1994.1258.

[23]

D. L. Wu, C. Li and P. F. Yuan, Periodic solutions for a class of second-order Hamiltonian systems of precribed energy, Electron. J. Qual. Theo., 77 (2015), 1-10. 

[24]

S. Q. Zhang, Multiple closed orbits of fixed energy for N-body-type problems with gravitational potentials, J. Math. Anal. Appl., 208 (1997), 462-475.  doi: 10.1006/jmaa.1997.5338.

[25]

S. Q. Zhang, Periodic solutions for some second order Hamiltonian systems, Nonlinearity, 22 (2009), 2141-2150.  doi: 10.1088/0951-7715/22/9/005.

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