Periodic solutions for -body problems with fixed centers
© Zhao et al.; licensee Springer 2013
Received: 26 January 2013
Accepted: 2 May 2013
Published: 17 May 2013
In this paper, we prove the existence of a new periodic solution for -body problems with fixed centers and strong-force potentials. In this model, N particles with equal masses are fixed at the vertices of a regular N-gon and the th particle is fixed at the center of the N-gon, the th particle winding around N particles.
MSC:34C15, 34C25, 70F10.
1 Introduction and main results
Specially, Gordon  proved the Keplerian elliptical orbits are the minimizers of Lagrangian action defined on the space for non-zero winding numbers.
In this paper, we use a variational minimizing method to look for a periodic solution for the th particle which winds around the ().
Definition 1.1 
When some point on C goes around the curve once, its image point will go around a number of times. This number is defined as the winding number of the curve C relative to the point p and is denoted by .
We have the following theorem.
2 The proof of Theorem 1.1
We recall the following famous lemmas, which we need to prove Theorem 1.1.
Lemma 2.1 
If , , , and there exists such that , then .
If in and , s.t. , , then .
Lemma 2.2 (Palais’s symmetry principle )
Let σ be an orthogonal representation of a finite or compact group G on a real Hilbert space H, and let be such that for , . Set . Then the critical point of f in is also a critical point of f in H.
Lemma 2.3 
If X is a reflexive Banach space, M is a weakly closed subset of X, and , is weakly lower semi-continuous and coercive, then f attains its infimum on M.
Lemma 2.4 (Poincare-Wirtinger inequality)
Let and , then . And the inequality takes the equality if and only if , .
We now prove Theorem 1.1.
If , then . Since , can be regarded as the square of an equivalent norm for , so it is weakly lower semi-continuous, so .
If , then by Lemma 2.1, , we have . So, . Hence f is w.l.s.c.
which contradicts the inequality in (2.2). By Lemma 2.2, is the critical point of f in ; therefore, is a non-collision periodic solution.
This completes the proof of Theorem 1.1. □
The authors sincerely thank the referees for their many helpful comments and suggestions and also express their sincere gratitude to Professor Zhang Shiqing for his discussions and corrections. This work is supported by NSF of China and Youth Fund of Mianyang Normal University.
- Euler M: De motu coproris ad duo centra virium fixa attracti. Nov. Commun. Acad. Sci. Imp. Petrop. 1766, 10: 207–242.Google Scholar
- Euler M: De motu coproris ad duo centra virium fixa attracti. Nov. Commun. Acad. Sci. Imp. Petrop. 1767, 11: 152–184.Google Scholar
- Euler M: Probleme un corps etant attire en raison reciproque quarree des distances vers deux points fixes donnes trouver les cas ou la courbe decrite par ce corps sera algebrique. Hist. Acad. R. Sci. Bell. Lett. Berlin 1767, 2: 228–249.Google Scholar
- Ambrosetti A, Coti Zelati V: Critical points with lack of compactness and applications to singular dynamical systems. Ann. Mat. Pura Appl. 1987, 149: 237–259. 10.1007/BF01773936MathSciNetView ArticleMATHGoogle Scholar
- Ambrosetti A, Coti Zelati V: Periodic Solutions for Singular Lagrangian Systems. Springer, Boston; 1993.View ArticleMATHGoogle Scholar
- Bahri A, Rabinowitz PH: A minimax method for a class of Hamiltonian systems with singular potentials. J. Funct. Anal. 1989, 82: 412–428. 10.1016/0022-1236(89)90078-5MathSciNetView ArticleMATHGoogle Scholar
- Benci V, Giannoni G: Periodic solutions of prescribed energy for a class of Hamiltonian system with singular potentials. J. Differ. Equ. 1989, 82: 60–70. 10.1016/0022-0396(89)90167-8MathSciNetView ArticleMATHGoogle Scholar
- Degiovanni M, Giannoni F: Dynamical systems with Newtonian type potentials. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 1988, 15: 467–494.MathSciNetMATHGoogle Scholar
- Gordon W: Conservative dynamical systems involving strong forces. Trans. Am. Math. Soc. 1975, 204: 113–135.View ArticleMathSciNetMATHGoogle Scholar
- Gordon W: A minimizing property of Keplerian orbits. Am. J. Math. 1977, 99: 961–971. doi:10.2307/2373993 10.2307/2373993View ArticleMathSciNetMATHGoogle Scholar
- Rabinowitz PH: A note on periodic solutions of prescribed energy for singular Hamiltonian systems. J. Comput. Appl. Math. 1994, 52: 147–154. 10.1016/0377-0427(94)90354-9MathSciNetView ArticleMATHGoogle Scholar
- Siegel C, Moser J: Lectures on Celestial Mechanics. Springer, Berlin; 1971.View ArticleMATHGoogle Scholar
- Wang XR, He S: Lagrangian actions on 3-body problems with two fixed centers. Bound. Value Probl. 2012., 2012: Article ID 28Google Scholar
- Tanaka K: A prescribed energy problem for a singular Hamiltonian system with weak force. J. Funct. Anal. 1993, 113: 351–390. 10.1006/jfan.1993.1054MathSciNetView ArticleMATHGoogle Scholar
- Tanaka K: A prescribed energy problem for conservative singular Hamiltonian system. Arch. Ration. Mech. Anal. 1994, 128: 127–164. 10.1007/BF00375024View ArticleMathSciNetMATHGoogle Scholar
- Zhang SQ: Multiple geometrically distinct closed noncollision orbits of fixed energy for N -body type problems with strong force potentials. Proc. Am. Math. Soc. 1996, 124: 3039–3046. 10.1090/S0002-9939-96-03751-3View ArticleMathSciNetMATHGoogle Scholar
- Palais R: The principle of symmetric criticality. Commun. Math. Phys. 1979, 69: 19–30. 10.1007/BF01941322MathSciNetView ArticleMATHGoogle Scholar
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