The 3-body problem in $ \mathbb{R}^4 $ has 24 dimensions and is invariant under translations and rotations. We do the full symplectic symmetry reduction and obtain a reduced Hamiltonian in local symplectic coordinates on a reduced phase space with 8 dimensions. The Hamiltonian depends on two parameters $ \mu_1 > \mu_2 \ge 0 $, related to the conserved angular momentum. The limit $ \mu_2 \to 0 $ corresponds to the 3-dimensional limit. We show that the reduced Hamiltonian has three relative equilibria that are local minima and hence Lyapunov stable when $ \mu_2 $ is sufficiently small. This proves the existence of balls of initial conditions of full dimension that do not contain any orbits that are unbounded.

中文翻译：

---

3体问题的对称约简 \ begin {document} $ \ mathbb {R} ^ 4 $ \ end {document}

$ \ mathbb {R} ^ 4 $中的3体问题具有24个维度，在平移和旋转下不变。我们进行了完全辛对称性约简，并在具有8维的缩小相空间上获得了局部辛坐标中的哈密顿量。哈密​​顿量取决于与守恒角动量有关的两个参数$ \ mu_1> \ mu_2 \ ge 0 $。极限$ \ mu_2 \ to 0 $对应于3维极限。我们证明，简化的哈密顿量具有三个相对平衡点，它们是局部最小值，因此当$ \ mu_2 $足够小时，Lyapunov稳定。这证明了存在全尺寸初始条件的球，这些球不包含任何无界的轨道。