发表机构
Capital Normal University; The University of Southern Mississippi; Hebei University of Economics and Business（首都师范大学; 南密西西比大学; 河北经贸大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究平面四体风筝中心构型的退化性，在受限子空间和全空间中分别刻画凸与凹情形的非退化性及退化曲线，并揭示全空间框架的必要性。
AI 中文摘要
基于我们先前提出的在全配置空间中研究中心构型的直接方法，我们研究了具有任意正质量的平面四体风筝中心构型在风筝对称受限子空间和全配置空间中的退化性。我们推导了雅可比行列式的一个因式分解，该分解极大地简化了后续分析。用于研究雅可比奇点的方法包括严格区间算术、Krawczyk算子、定量隐函数定理以及计算机辅助证明。我们同时考虑了凸和凹的风筝中心构型。在凸情形下，我们证明了在受限子空间和全配置空间中均具有非退化性。在凹情形下，受限子空间中存在一条唯一的光滑退化曲线；在全空间设置中出现了另一条退化曲线，该曲线在风筝构型子空间中被隐藏，这使得在全配置空间中研究退化性成为不可避免的。我们还建立了这些退化曲线的若干性质。作为这一完整刻画的一个例证，我们检验了著名的等边三角形中心构型，其中顶点处质量相等且第四个质量位于中心，这揭示了我们的全空间框架所捕捉到的退化性的额外特征。
英文摘要
Based on our previously proposed direct approach to central configurations in the full configuration space, we study the degeneracy of planar four-body kite central configurations with arbitrary positive masses both in kite symmetry restricted subspace and in full configuration space. We derive a factorization of the determinant of the Jacobian that substantially simplifies the subsequent analysis. The methods used to study the singularity of the Jacobian include rigorous interval arithmetic, the Krawczyk operator, a quantitative implicit function theorem, and computer-assisted proofs. Both convex and concave kite central configurations are considered. In the convex case, we prove the nondegeneracy in both the restricted subspace and the full configuration space. In the concave case, there is a unique smooth degeneracy curve in the restricted subspace; an additional degeneracy curve emerges in the full space setting, which is hidden in the subspace of kite configurations and makes it unavoidable to study the degeneracy in the full configuration space. We also establish several properties of these degeneracy curves. As an illustration to this complete characterization, we exam the well-known central configuration of equilateral triangle with equal masses at the vertices and a fourth mass at the center, which reveals additional features of degeneracy captured by our full-space framework.
Comments43 pages, 11 figures
