We consider the dynamics and symplectic reduction of the 2-body problem on a sphere of arbitrary dimension. It suffices to consider the case when the sphere is 3-dimensional. As the 3-sphere is a group it acts on itself by left and right multiplication and these together generate the action of the SO(4) symmetry on the sphere. This gives rise to a notion of left and right momenta for the problem, and allows for a reduction in stages, first by the left and then the right, or vice versa. The intermediate reduced spaces obtained by left or right reduction are shown to be coadjoint orbits of the special Euclidean group SE(4). The full reduced spaces are generically 4-dimensional and we describe these spaces and their singular strata. The dynamics of the 2-body problem descend through a double cover to give a dynamical system on SO(4) which, after reduction and for a particular choice of Hamiltonian, coincides with that of a 4-dimensional spinning top with symmetry. This connection allows us to “hit two birds with one stone” and derive results about both the spinning top and the 2-body problem simultaneously. We provide the equations of motion on the reduced spaces and fully classify the relative equilibria and discuss their stability.

中文翻译：

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3球体和4维旋转陀螺上的2体问题的奇异约化

我们考虑任意尺寸球体上2体问题的动力学和辛简化。只需考虑球体为3维时的情况即可。由于3球是一个组，因此它通过左右乘法作用于自身，它们一起在球上产生SO（4）对称性的作用。这引起了该问题的左右动量的概念，并允许减少阶段，首先是减少，然后是减少，反之亦然。通过左或右缩减获得的中间缩减空间显示为特殊欧几里得群SE的共伴轨道（4）。完整的缩减空间一般是4维的，我们将描述这些空间及其奇异层。2体问题的动力学通过双层覆盖下降，从而在SO（4）上给出了一个动力学系统，该动力学系统经过还原并针对特定的哈密顿量进行选择，与对称的4维旋转陀螺相吻合。这种联系使我们可以“用一块石头砸两只鸟”，并同时得出有关陀螺和两体问题的结果。我们提供了缩减空间上的运动方程，并对相对平衡进行了完全分类，并讨论了它们的稳定性。