We consider the 2-body problem in the sphere \(\mathbb{S}^{2}\). This problem is modeled by a Hamiltonian system with \(4\) degrees of freedom and, following the approach given in [4], allows us to reduce the study to a system of \(2\) degrees of freedom. In this work we will use theoretical tools such as normal forms and some nonlinear stability results on Hamiltonian systems for demonstrating a series of results that will correspond to the open problems proposed in [4] related to the nonlinear stability of the relative equilibria. Moreover, we study the existence of Hamiltonian pitchfork and center-saddle bifurcations.

中文翻译：

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球面上二体问题中相对平衡的稳定性

我们考虑球体\(\mathbb{S}^{2}\) 中的二体问题。这个问题由一个具有\(4\)自由度的哈密​​顿系统建模，并且按照 [4] 中给出的方法，允许我们将研究简化为\(2\)自由度的系统。在这项工作中，我们将使用理论工具，例如标准形式和哈密顿系统上的一些非线性稳定性结果，以展示一系列结果，这些结果将对应于 [4] 中提出的与相对平衡的非线性稳定性相关的开放问题。此外，我们研究了哈密顿干草叉和中心鞍形分岔的存在。