论文标题
牛仔$ n $ n $ body问题中的$ s $平衡配置的摩尔斯理论
Morse theory for $S$-balanced configurations in the Newtonian $n$-body problem
论文作者
论文摘要
对于北部（引力）$ n $ - 体育问题，欧几里得$ d $维空间，最简单的解决方案是由那些僵化的动作（同型解决方案）提供的，在这种方案中，每个身体都沿着开plerian轨道移动，并且$ n $ n $ n $的配置是n $ n $ - body的配置，即持续不断地提高旋转和尺度的旋转和缩放\ textIt \ textIt \ textIt \ textIt \ textIt}。对于$ d \ leq 3 $，唯一可能的同型动作是中央配置给出的动作。对于Albouy和Chenciner所观察到的正交组$ O（D）$的复杂性较高，因此出现了$ d \ geq 4 $，而出现了新的可能性。例如，在$ \ mathbb r^4 $中，可以在两个具有不同角速度的相互正交平面上旋转。这在引力和离心力之间产生了新的平衡，从而提供新的周期性和准周期性动作。因此，对于$ d \ geq 4 $，$ s $ - \ textit {balanced配置}（包含中心的）提供了一个简单的解决方案，可以通过关键点理论来表征$ n $ body问题的简单解决方案。在本文中，我们首先在$ \ mathbb r^d $中的平衡（非中心）配置的数量，用于任意$ d \ geq 4 $中，并建立一个$ 45^\ circ $ - circ $ - theorem的版本，以用于平衡配置，从而回答了Moeckel提出的一些问题。同样，对无碰撞配置球的庞加莱多项式系数的渐近学仔细研究将使我们能够在$ s $平衡的配置计数中得出一些相当出乎意料的定性后果。在本文的最后一部分中，我们重点介绍$ d = 4 $，并在引力$ n $ n $ body问题的定期和准周期性动作数量上提供了下限，从而改善了先前的麦克科德（McCord）著名结果。
For the Newtonian (gravitational) $n$-body problem in the Euclidean $d$-dimensional space, the simplest possible solutions are provided by those rigid motions (homographic solutions) in which each body moves along a Keplerian orbit and the configuration of the $n$-body is a constant up to rotations and scalings named \textit{central configuration}. For $d\leq 3$, the only possible homographic motions are those given by central configurations. For $d \geq 4$ instead, new possibilities arise due to the higher complexity of the orthogonal group $O(d)$, as observed by Albouy and Chenciner. For instance, in $\mathbb R^4$ it is possible to rotate in two mutually orthogonal planes with different angular velocities. This produces a new balance between gravitational forces and centrifugal forces providing new periodic and quasi-periodic motions. So, for $d\geq 4$ there is a wider class of $S$-\textit{balanced configurations} (containing the central ones) providing simple solutions of the $n$-body problem, which can be characterized as well through critical point theory. In this paper, we first provide a lower bound on the number of balanced (non-central) configurations in $\mathbb R^d$, for arbitrary $d\geq 4$, and establish a version of the $45^\circ$-theorem for balanced configurations, thus answering some questions raised by Moeckel. Also, a careful study of the asymptotics of the coefficients of the Poincaré polynomial of the collision free configuration sphere will enable us to derive some rather unexpected qualitative consequences on the count of $S$-balanced configurations. In the last part of the paper, we focus on the case $d=4$ and provide a lower bound on the number of periodic and quasi-periodic motions of the gravitational $n$-body problem which improves a previous celebrated result of McCord.
