弯曲n体问题的一种内禀方法：负曲率情形
An intrinsic approach in the curved n-body problem: the negative curvature case
摘要：
研究了具有负常高斯曲率的二维双曲球上正质量的n个点粒子在引力作用下的运动。利用球极平面投影，我们导出了Poincaré圆盘中的曲n体问题的运动方程，并研究了椭圆相对平衡点。为了分析双曲型和抛物型的相对平衡，我们得到了Poincaré上半平面中的运动方程。利用黎曼几何的技巧，我们刻画了上述每一类周期轨道。对于n=2和n=3，我们恢复一些以前已知的结果，并找到新的定性结果的相对平衡，不明显的外在设置。
英文摘要：
We consider the motion of n point particles of positive masses that interact gravitationally on the 2-dimensional hyperbolic sphere, which has negative constant Gaussian curvature. Using the stereographic projection, we derive the equations of motion of this curved n-body problem in the Poincaré disk, where we study the elliptic relative equilibria. Then we obtain the equations of motion in the Poincaré upper half plane in order to analyze the hyperbolic and parabolic relative equilibria. Using techniques of Riemannian geometry, we characterize each of the above classes of periodic orbits. For n=2 and n=3 we recover some previously known results and find new qualitative results about relative equilibria that were not apparent in an extrinsic setting.
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