Periodic solutions for N + 2 -body problems with N + 1 fixed centers

Furong Zhao12*, Fengying Li2 and Jian Chen23

Author Affiliations

1 Department of Mathematics and Computer Science, Mianyang Normal University, Mianyang, Sichuan, 621000, P.R. China

2 Department of Mathematics, Sichuan University, Chengdu, 610064, P.R. China

3 School of Science, Southwest University of Science and Technology, Mianyang, Sichuan, 621000, P.R. China

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Boundary Value Problems 2013, 2013:129  doi:10.1186/1687-2770-2013-129

 Received: 26 January 2013 Accepted: 2 May 2013 Published: 17 May 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

Keywords:
-body problems with -fixed centers; minimizing variational methods; strong-force potentials

1 Introduction and main results

In the eighteenth century, the 2-fixed center problem was studied by Euler [1-3]. 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:

(1.1)

Equivalently,

(1.2)

(1.3)

where

The type of system (1.2) is called a singular Hamiltonian system which attracts many researchers (see [1-10] and [11-16]).

Specially, Gordon [10] 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[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

(1.4)

(1.5)

(1.6)

(1.7)

(1.8)

We have the following theorem.

Theorem 1.1For, the minimizer ofon () exists and it is a non-collision periodic solution of (1.1) or (1.2)-(1.3) (please see Figures1-4for).

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.

Ifinand, s.t. , , 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 semi-continuous and coercive, thenfattains its infimum onM.

Lemma 2.4 (Poincare-Wirtinger 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 ,

(2.1)

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 semi-continuous, 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

(2.2)

If is a collision periodic solution, then there exist and such that . Let and note . By Lemma 2.1, we have

(2.3)

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. □

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.

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