论文标题
$ \ mathbb {r}^4 $中三体问题的相对平衡
Relative equilibria of the 3-body problem in $\mathbb{R}^4$
论文作者
论文摘要
牛顿三体问题的经典方程不仅定义了熟悉的3维动作。运动的尺寸也可能是4，并且不能更高。我们证明，在维度4中，对于三个任意的正质量，对于角动量的任意值（等级4），该能量具有最小值，这对应于相对平衡的运动，当将Lyapunov稳定时，当将其视为减少问题的平衡时。附近的动作是非词性的，并且一直有限。我们还描述了相对平衡的完整家族，并表明其图像通过能量摩托车地图呈现出尖和其他有趣的特征。
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.
