Abstract
In this paper, we prove the existence of a new periodic solution for body problems with fixed centers and strongforce potentials. In this model, N particles with equal masses are fixed at the vertices of a regular Ngon and the th particle is fixed at the center of the Ngon, the th particle winding around N particles.
MSC: 34C15, 34C25, 70F10.
Keywords:
body problems with fixed centers; minimizing variational methods; strongforce potentials1 Introduction and main results
In the eighteenth century, the 2fixed center problem was studied by Euler [13]. Here, let us consider the fixed center problem: We assume N particles with equal masses 1 are fixed at the vertices () of a regular polygon and the th particle is fixed at the origin , the th particle with mass is attracted by the other particles, and moves according to Newton’s second law and a more general power law than the Newton’s universal gravitational square law. In this system, the position for the th particle satisfies the following equation:
Equivalently,
where
The type of system (1.2) is called a singular Hamiltonian system which attracts many researchers (see [110] and [1116]).
Specially, Gordon [10] proved the Keplerian elliptical orbits are the minimizers of Lagrangian action defined on the space for nonzero 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[10]
Let be a given oriented closed curve, and . Define :
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 .
Let
We have the following theorem.
Theorem 1.1For, the minimizer ofon () exists and it is a noncollision periodic solution of (1.1) or (1.2)(1.3) (please see Figures14for).
2 The proof of Theorem 1.1
We recall the following famous lemmas, which we need to prove Theorem 1.1.
Lemma 2.1[9]
If, , , and there existssuch that, then.
Lemma 2.2 (Palais’s symmetry principle [17])
Letσbe an orthogonal representation of a finite or compact groupGon a real Hilbert spaceH, and letbe such that for, . Set. Then the critical point offinis also a critical point offinH.
Lemma 2.3[5]
IfXis a reflexive Banach space, Mis a weakly closed subset ofX, and, is weakly lower semicontinuous and coercive, thenfattains its infimum onM.
Lemma 2.4 (PoincareWirtinger inequality)
Letand, then. And the inequality takes the equality if and only if, .
We now prove Theorem 1.1.
Proof By the symmetry of , we know for ,
If in , then by Sobolev’s compact embedding theorem, we have in .
(i) If , then . Since , can be regarded as the square of an equivalent norm for , so it is weakly lower semicontinuous, so .
(ii) If , then by Lemma 2.1, , we have . So, . Hence f is w.l.s.c.
Using (2.1), we know that is coercive on . Lemma 2.3 guarantees that attains its infimum on . Let the minimizer be , then
If is a collision periodic solution, then there exist and such that . Let and note . By Lemma 2.1, we have
which contradicts the inequality in (2.2). By Lemma 2.2, is the critical point of f in ; therefore, is a noncollision periodic solution.
This completes the proof of Theorem 1.1. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read, checked and approved the final manuscript.
Acknowledgements
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.
References

Euler, M: De motu coproris ad duo centra virium fixa attracti. Nov. Commun. Acad. Sci. Imp. Petrop.. 10, 207–242 (1766)

Euler, M: De motu coproris ad duo centra virium fixa attracti. Nov. Commun. Acad. Sci. Imp. Petrop.. 11, 152–184 (1767)

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. 2, 228–249 (1767)

Ambrosetti, A, Coti Zelati, V: Critical points with lack of compactness and applications to singular dynamical systems. Ann. Mat. Pura Appl.. 149, 237–259 (1987). Publisher Full Text

Ambrosetti, A, Coti Zelati, V: Periodic Solutions for Singular Lagrangian Systems, Springer, Boston (1993)

Bahri, A, Rabinowitz, PH: A minimax method for a class of Hamiltonian systems with singular potentials. J. Funct. Anal.. 82, 412–428 (1989). Publisher Full Text

Benci, V, Giannoni, G: Periodic solutions of prescribed energy for a class of Hamiltonian system with singular potentials. J. Differ. Equ.. 82, 60–70 (1989). Publisher Full Text

Degiovanni, M, Giannoni, F: Dynamical systems with Newtonian type potentials. Ann. Sc. Norm. Super. Pisa, Cl. Sci.. 15, 467–494 (1988)

Gordon, W: Conservative dynamical systems involving strong forces. Trans. Am. Math. Soc.. 204, 113–135 (1975)

Gordon, W: A minimizing property of Keplerian orbits. Am. J. Math.. 99, 961–971 doi:10.2307/2373993 (1977)
doi:10.2307/2373993
Publisher Full Text 
Rabinowitz, PH: A note on periodic solutions of prescribed energy for singular Hamiltonian systems. J. Comput. Appl. Math.. 52, 147–154 (1994). Publisher Full Text

Siegel, C, Moser, J: Lectures on Celestial Mechanics, Springer, Berlin (1971)

Wang, XR, He, S: Lagrangian actions on 3body problems with two fixed centers. Bound. Value Probl.. 2012, Article ID 28 (2012)

Tanaka, K: A prescribed energy problem for a singular Hamiltonian system with weak force. J. Funct. Anal.. 113, 351–390 (1993). Publisher Full Text

Tanaka, K: A prescribed energy problem for conservative singular Hamiltonian system. Arch. Ration. Mech. Anal.. 128, 127–164 (1994). Publisher Full Text

Zhang, SQ: Multiple geometrically distinct closed noncollision orbits of fixed energy for Nbody type problems with strong force potentials. Proc. Am. Math. Soc.. 124, 3039–3046 (1996). Publisher Full Text

Palais, R: The principle of symmetric criticality. Commun. Math. Phys.. 69, 19–30 (1979). Publisher Full Text