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:
Specially, Gordon  proved the Keplerian elliptical orbits are the minimizers of Lagrangian action defined on the space for non-zero winding numbers.
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 2.2 (Palais’s symmetry principle )
Lemma 2.4 (Poincare-Wirtinger inequality)
We now prove Theorem 1.1.
This completes the proof of Theorem 1.1. □
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read, checked and approved the final manuscript.
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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