论文标题
牛顿$ n $ body问题的平衡配置的分叉中的$ \ mathbb r^4 $
Bifurcations of balanced configurations for the Newtonian $n$-body problem in $\mathbb R^4$
论文作者
论文摘要
对于重力$ n $ - 体问题，最简单的动作是由每个身体沿开普勒轨道移动的那些刚体动作提供的，系统的形状是具有合适属性的常数（直至旋转和尺度）配置。虽然在尺寸$ d \ leq 3 $中，配置必须是核心，但在尺寸$ d \ geq 4 $中，由于正交组的复杂性而产生了新的可能性，并且实际上，$ s $平衡的配置包含$ n $ n $ - $ n $ body问题的简单解决方案。从第一和第三作者的最新结果开始，我们研究了从共线$ s $平衡的配置的琐碎分支分支的连续分支的存在，并从下面提供了关于分叉瞬间数量的估计。在本文的最后一部分中，通过使用持续方法，在三个身体问题的情况下，我们明确显示分叉分支，以实现不同的质量选择。
For the gravitational $n$-body problem, the simplest motions are provided by those rigid motions in which each body moves along a Keplerian orbit and the shape of the system is a constant (up to rotations and scalings) configuration featuring suitable properties. While in dimension $d \leq 3$ the configuration must be central, in dimension $d \geq 4$ new possibilities arise due to the complexity of the orthogonal group, and indeed there is a wider class of $S$-balanced configurations, containing central ones, which yield simple solutions of the $n$-body problem. Starting from recent results of the first and third authors, we study the existence of continua of bifurcations branching from a trivial branch of collinear $S$-balanced configurations and provide an estimate from below on the number of bifurcation instants. In the last part of the paper, by using the continuation method, we explicitly display the bifurcation branches in the case of the three body problem for different choices of the masses.
