angular momentum为零的三体运动的整体几何，I
Global geometry of 3-body motions with vanishing angular momentum, I
摘要：
根据Lagrange的最小作用量原理的Jacobi几何化，经典力学的轨线可以表示为位形空间M上关于一个适当度量的测地线，该度量是运动学度量的一个因子（U+h）的保角修正，其中U和h分别是势函数和总能量。这种轨迹的全局几何形状可以简化为它们的模曲线的几何形状，模曲线记录了定向m-三角形的模空间中的大小和形状的变化，定向m-三角形的运动学度量实际上是形状空间M*（半径为1/2的2-球面）上的黎曼锥。在本文中，我们表明，这样的运动的模量曲线是唯一确定的形状曲线（只记录形状的变化）时，h是非零的，而在特殊情况下，h=0，它是唯一确定的标度。因此，对这种运动的全局几何的研究可以进一步简化为形状曲线的研究，形状曲线是由三阶常微分方程表征的2-球面上的时间参数化曲线（参见图1）。定理3.9）。此外，这些曲线有两个显着的性质，即参数化的唯一性和单调性（参见。定理4.6和定理5.8），构成了一个坚实的基础，系统地研究其整体几何，并自然导致制定一些相关的问题，如那些简要讨论的第7节。
英文摘要：
Following Jacobi's geometrization of Lagrange's least action principle, trajectories of classical mechanics can be characterized as geodesics on the configuration space M with respect to a suitable metric which is the conformal modification of the kinematic metric by a factor (U+h), where U and h is the potential function and total energy, respectively.In the special case of 3-body motions with zero angular momentum, the global geometry of such trajectories can be reduced to that of their moduli curves, which record the change of size and shape, in the moduli space of oriented m-triangles, whose kinematic metric is, in fact, a Riemannian cone over the shape space M*(a 2-sphere of radius 1/2). In this paper, we show that the moduli curve of such a motion is uniquely determined by its shape curve (which only records the change of shape)when h is nonzero, while in the special case of h=0 it is uniquely determined up to scaling. Thus, the study of the global geometry of such motions can be further reduced to that of the shape curves, which are time-parametrized curves on the 2-sphere characterized by a third order ODE (cf. Theorem 3.9). Moreover, these curves have two remarkable properties, namely the uniqueness of parametrization and the monotonicity (cf. Theorem 4.6 and Theorem 5.8) that constitute a solid foundation for a systematic study of their global geometry and naturally lead to the formulation of some pertinent problems such as those briefly discussed in Section 7.
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