Abstract
In this paper, we study solution structures of the following generalized Lennard-Jones system in ℝn,
, with 0 < α < β. Considering periodic solutions with zero angular momentum, we prove that the corresponding problem degenerates to 1-dimensional and possesses infinitely many periodic solutions which must be oscillating line solutions or constant solutions. Considering solutions with non-zero angular momentum, we compute Morse indices of the circular solutions first, and then apply the mountain pass theorem to show the existence of non-circular solutions with non-zero topological degrees. We further prove that besides circular solutions the system possesses in fact countably many periodic solutions with arbitrarily large topological degree, infinitely many quasi-periodic solutions, and infinitely many asymptotic solutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ambrosetti, A., Coti Zelati, V.: Periodic Solutions of Singular Lagrangian Systems. Progr. Nonlinear Differential Equations Appl., 10, Birkhäuser, Boston, 1993
Arnold, V. I.: Mathematical Methods of Classical Mechanics, Springer, Berlind, 1978
Bahri, A., Rabinowitz, P.: A minimax method for a class of Hamiltonian systems with singular potentials. J. Funct. Anal., 82, 412–428 (1989)
Bărbosu, M., Mioc, V., Paşca, D., et al.: The two-body problem with generalized Lennard-Jones potential. J. Math. Chem., 49, 1961–1975 (2011)
Brush, S. G.: Interatomic forces and gas theory from newton to Lennard-Jones. Arch. Ration. Mech. Anal., 39, 1–29 (1970)
Chang, K. C.: Infinite-dimensional Morse Theory and Multiple Solution Problems. Prog. in Nonlinear Diff. Equa. and their Appl., 6, Birkhäuser Boston, Inc., Boston, 1993
Corbera, M., Llibre, J., Pérez-Chavela, E.: Equilibrium points and central configurations for the Lennard- Jones 2- and 3-body problems. Cele. Mech. Dynam. Astro., 89, 235–266 (2004)
Corbera, M., Llibre, J., Pérez-Chavela, E.: Symmetric planar non-collinear relative equilibria for the Lennard-Jones potential 3-body problem with two equal masses. Proceedings of the 6th Conf. on Cele. Mech. (Spanish), Monogr. Real Acad. Ci. Exact. F’is.-Quí m. Nat. Zaragoza, 25, pp.93–114. Real Acad. Ci. Exact., Fís. Quím. Nat. Zar, Zaragoza, 2004
Coti-Zelati, V.: Dynamical systems with effective-like potentials. Nonlinear Anal., 12, 209–222 (1988)
Gordon, W. B.: Conservative dynamical systems involving strong forces. Trans. Amer. Math. Soc., 204, 113–135 (1975)
Hofer, H.: A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem. J. London Math. Soc., 31, 566–570 (1985)
Komornik, V.: Another short proof of Descartes’s rule of signs. Amer. Math. Monthly, 113, 829–830 (2006)
Jones, J. E.: On the determination of molecular fields. II. from the equation of state of a gas. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 106, 463–477 (1924)
Liu, B.: Dortoral Thesis, Nankai University, in preparation
Landau, L. D., Lifshitz, E. M.: Mechanics. Course of Theoretical Physics, Vol. 1. Translated from Russian by J. B. Bell. Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., 1960
Llibre, J., Long, Y.: Periodic solutions for the generalized anisotropic Lennard-Jones Hamiltonian. Qual. Theory Dyn. Syst., 14, 291–311 (2015)
Long, Y.: Multiple solutions of perturbed superquadratic second order Hamiltonian systems. Trans. Amer. Math. Soc., 311, 749–780 (1989)
Paşca, D., Valls, C.: Qualitative analysis of the anisotropic two-body problem with generalized Lennard- Jones potential. J. Math. Chem., 50, 2671–2688 (2012)
Rabinowitz, P. H.: Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math., 31, 157–184 (1978)
Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol.65 CBMS Reg. Conf. Series in Math. Amer. Math. Soc., Providence, RI, 1986
Sbano, L., Southall, J.: Periodic solutions of the N-body problem with Lennard-Jones-type potentials. Dyn. Syst., 25, 53–73 (2010)
Solimini, S.: On forced dynamical systems with a singularity of repulsive type. Nonlinear Anal., 14, 489–500 (1990)
Terracini, S.: Remarks on periodic orbits of dynamical systems with repulsive singularities. J. Funct. Anal., 111, 213–238 (1993)
Tian, G.: On the mountain-pass lemma. Kexue Tongbao (English Ed.), 29, 1150–1154 (1984)
Acknowledgements
We thank the referees for their careful reading and pointing out some typos in the paper. Yiming Long would like to express his sincere thanks to Prof. Jaume Llibre for valuable discussions on this topic.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is partially supported by the Ph.D. Candidate Research Innovation Fund of Nankai University and NSFC (Grant Nos. 11131004 and 11671215); the second author is partially supported by NSFC (Grant Nos. 11131004 and 11671215), LPMC of MOE of China, Nankai University, and the BAICIT at Capital Normal University; and the third author is partially supported by US NSF (Grant DMS-1362507)
Rights and permissions
About this article
Cite this article
Liu, B., Long, Y. & Zeng, C. Solutions of the generalized Lennard-Jones system. Acta. Math. Sin.-English Ser. 34, 139–170 (2018). https://doi.org/10.1007/s10114-017-7139-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-017-7139-6
Keywords
- Generalized Lennard-Jones system
- mountain pass solutions
- periodic solutions
- quasiperiodic solutions
- asymptotic solutions