发表机构
Zhejiang University; Universität Augsburg（浙江大学; 奥格斯堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究对带电N体问题的静态平衡进行分类，证明三体平衡均共线且不稳定，并给出四体及共线平衡的结式条件与逆实现定理，最后分析正多边形构型。
AI 中文摘要
我们研究带电$N$体问题中的非碰撞静态平衡。与经典牛顿情形不同，此类平衡可能存在，因为有效相互作用系数$\delta_{ij}=m_im_j-e_ie_j$可以为正、负或零。我们给出了三体平衡的完整分类，并证明所有三体平衡都是共线的且线性不稳定。对于四体问题，我们推导了凸和非凸非共线平衡的符号限制，排除了非平凡的共圆四边形平衡，并通过辅助三角形和结式方程获得了一个几何必要条件。我们还通过将力平衡方程化简为齐次多项式系统来研究共线平衡。这产生了一个结式必要条件和一个逆实现定理，表明任意给定的由不同点组成的共线构型都可以通过适当的正质量和实电荷实现。最后，我们分析了正多边形构型和中心化正多边形构型，特别证明了等质量不能形成非平凡的$N$边形正多边形平衡，而中心化情形则归结为单个标量条件。
英文摘要
We study non-collision static equilibria in the charged \(N\)-body problem. Unlike the classical Newtonian case, such equilibria may exist because the effective interaction coefficients $δ_{ij}=m_im_j-e_ie_j$ can be positive, negative, or zero. We give a complete classification of three-body equilibria and prove that all of them are collinear and linearly unstable. For the four-body problem, we derive sign restrictions for convex and concave non-collinear equilibria, exclude non-trivial concyclic quadrilateral equilibria, and obtain a geometric necessary condition expressed through an auxiliary triangle and a resultant equation. We also study collinear equilibria by reducing the force-balance equations to a homogeneous polynomial system. This yields a resultant necessary condition and an inverse realization theorem showing that every prescribed collinear configuration of distinct points can be realized by suitable positive masses and real charges. Finally, we analyze regular polygon configurations and centered regular polygon configurations, showing in particular that equal masses cannot form a non-trivial regular \(N\)-gon equilibrium, while the centered case reduces to a single scalar condition.
Comments57 pages, 4 figures
