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) central configuration. For \(d\le 3\), the only possible homographic motions are those given by central configurations. For \(d \ge 4\) instead, new possibilities arise due to the higher complexity of the orthogonal group \(\mathrm {O}(d)\), as observed by Albouy and Chenciner (Invent Math 131(1):151–184, 1998). 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 \(d\ge 4\) there is a wider class of S -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 \(d\ge 4\), and establish a version of the \(45^\circ \)-theorem for balanced configurations, thus answering some of the questions raised in Moeckel (Central configurations, 2014). 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 (Ergodic Theory Dyn Syst 16:1059–1070, 1996).

对于欧几里得d维空间中的牛顿（引力）n体问题，最简单的可能解决方案是由那些刚性运动（单应解决方案）提供的，其中每个物体都沿着开普勒轨道运动，n体的配置是a（恒定到旋转和缩放）中央配置。对于\(d\le 3\)，唯一可能的单应运动是由中心配置给出的运动。对于\(d \ge 4\) 而言，由于正交群\(\mathrm {O}(d)\)的更高复杂性，出现了新的可能性，正如 Albouy 和 Chenciner（发明数学 131(1) ）所观察到的： 151-184，1998 年）。例如，在\(\mathbb {R}^4\)可以在两个相互正交的平面内以不同的角速度旋转。这在重力和离心力之间产生了新的平衡，提供了新的周期性和准周期性运动。因此，对于\(d\ge 4\)有更广泛的S 平衡配置（包含中心配置）提供n体问题的简单解决方案，也可以通过临界点理论对其进行表征。在本文中，我们首先提供\(\mathbb {R}^d\)中平衡（非中心）配置数量的下限，对于任意\(d\ge 4\)，并建立一个版本的\（45 ^ \ CIRC \）- 平衡配置定理，从而回答了 Moeckel（中央配置，2014 年）中提出的一些问题。此外，仔细研究无碰撞构型球的庞加莱多项式系数的渐近性将使我们能够推导出一些相当出乎意料的关于S平衡构型计数的定性结果。在论文的最后一部分，我们专注于\(d=4\) 的情况，并提供了引力n体问题的周期和准周期运动次数的下界，这改进了之前 McCord 的著名结果（遍历理论 Dyn Syst 16：1059-1070，1996）。