The \(2\)-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions concerning the hyperbolic system by complexifying it and treating it as the complexification of a spherical system. In this way, results for the \(2\)-body problem on the sphere are readily translated to the hyperbolic case. For instance, we implement this idea to completely classify the relative equilibria for the \(2\)-body problem on hyperbolic 3-space and discuss their stability for a strictly attractive potential.

中文翻译：

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用双四元数统一双曲和球面$$2$$-体问题

球面和双曲空间上的\(2\)体问题都是复球面上定义的全纯哈密顿系统的实数形式。这允许用双四元数进行自然描述，并允许我们通过复杂化双曲系统并将其视为球面系统的复杂化来解决有关双曲系统的问题。这样，球体上的\(2\)体问题的结果很容易转换为双曲线情况。例如，我们实现这个想法来对双曲 3 空间上的2 体问题的相对平衡进行完全分类，并讨论它们对于严格吸引势的稳定性。