The classical equations of the Newtonian 3-body problem do not only define the familiar 3-dimensional motions. The dimension of the motion may also be 4, and cannot be higher. We prove that in dimension 4, for three arbitrary positive masses, and for an arbitrary value (of rank 4) of the angular momentum, the energy possesses a minimum, which corresponds to a motion of relative equilibrium which is Lyapunov stable when considered as an equilibrium of the reduced problem. The nearby motions are nonsingular and bounded for all time. We also describe the full family of relative equilibria, and show that its image by the energy-momentum map presents cusps and other interesting features.

中文翻译：

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$ \ mathbb {R} ^ 4 $中三体问题的相对平衡

牛顿3体问题的经典方程式不仅定义了熟悉的3维运动。运动的尺寸也可以是4，并且不能更高。我们证明，在维度4中，对于三个任意正质量，对于任意角动量值（秩4），该能量都具有最小值，该能量对应于相对平衡运动，当视为相对平衡运动时，李雅普诺夫稳定。简化问题的均衡。附近的运动始终是非奇异的。我们还描述了相对平衡的整个家族，并通过能量动量图显示其图像具有尖峰和其他有趣的特征。