A continuum of periodic solutions to the planar four-body problem with two pairs of equal masses

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:264

Article

Ouyang, Tiancheng¹  Xie, Zhifu² 

[1] Brigham Young Univ, Dept Math, Provo, UT 84602 USA

[2] Univ Southern Mississippi, Dept Math, Hattiesburg, MS 39406 USA

关键词: Variational method;    Choreographic periodic solutions;    Structural Prescribed Boundary Conditions (SPBC);    Stability;    Central configurations;    n-body problem;   

DOI  :  10.1016/j.jde.2017.12.016

来源: Elsevier

【 摘 要 】

In this paper, we apply the variational method with Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for the planar four-body problem with two pairs of equal masses m(1)=m(3) and m(2) . A path q(t) on [0,T] satisfies the SPBC if the boundaries q(0) is an element of A and q(T) is an element of B, where A and B are two structural configuration spaces in (R-2)(4) and they depend on a rotation angle theta is an element of(0, 2 pi) and the mass ratio mu = m(2)/m(1) is an element of R+. We show that there is a region Omega subset of (0, 2 pi) x R+ such that there exists at least one local minimizer of the Lagrangian action functional on the path space satisfying the SPBC {q(t) is an element of H-1 ([0,T], (R-2)(4)) vertical bar q(0) is an element of A, q (T) is an element of B} for any (theta, mu) is an element of Omega . The corresponding minimizing path of the minimizer can be extended to a non-homographic periodic solution if theta is commensurable with pi or a quasi-periodic solution if theta is not commensurable with pi. In the variational method with the SPBC, we only impose constraints on the boundary and we do not impose any symmetry constraint on solutions. Instead, we prove that our solutions that are extended from the initial minimizing paths possess certain symmetries. The periodic solutions can be further classified as simple choreographic solutions, double choreographic solutions and non-choreographic solutions. Among the many stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution when (theta, mu) = (4 pi/5, 1). Remarkably the unequal-mass variants of the stable star pentagon are just as stable as the equal mass choreographies. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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