对称$N$体问题中周期与拟周期轨道的变分持续性

中文摘要 AI 辅助

许多力学系统，如牛顿$n$体系统，在旋转下保持不变。系统的旋转对称性产生了两个交换作用：动力学作用和对称群的作用。在本文中，我们利用这种对称性来构造周期和拟周期轨迹族。在强力设定下，我们将Montgomery的变分方法推广到自由同伦类中那些在给定非平凡旋转下周期性的曲线。在弱力设定下，我们在适当假设下，对于由一般有限对称群作用在足够小旋转下定义的环空间中的作用极小元，建立了全局和局部的变分持续性结果。作为应用，我们将这些结果应用于八字轨道（在数值支持的严格局部极小性假设下）、具有$q$重旋转对称性的双编舞环，以及平面$(N+3)$体问题中的反向旋转双编舞环。

英文摘要

Many mechanical systems, such as the Newtonian $n$-body system, are invariant under rotations. The rotational symmetry of the system gives rise to two commuting actions: the dynamical action and the action of the symmetry group. In this paper, we exploit this symmetry to construct families of periodic and quasi-periodic trajectories. In the strong-force setting, we extend Montgomery's variational approach to free homotopy classes of curves that are periodic up to a prescribed nontrivial rotation. In the weak-force setting, we establish, under suitable hypotheses, global and local variational persistence results for action minimizers in loop spaces defined by general finite symmetry-group actions under sufficiently small rotations. As applications, we apply these results to the figure-eight orbit, under a numerically supported assumption of strict local minimality, to double choreographic loops with $q$-fold rotation symmetry, and to counter-rotating double choreographic loops in planar $(N+3)$-body problems.

发表机构

• Rice University（莱斯大学）
• Northwestern University（西北大学）
• Nankai University（南开大学）

机构由 AI 辅助整理，请以论文原文为准。