平面三体问题几何方法中的曲率和测地线不稳定性
Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem
摘要：
莫佩屠斯原理允许我们把经典的轨迹重新参数化测地线的雅可比莫佩屠斯（JM）度量的配置空间。我们研究这测地线重新与牛顿和有吸引力的平方反比潜力的平面三体问题。相关的JM度规除了具有零能量E的平方反比势的标度等距外，还具有平移和旋转等距。全位形空间$C^3$（不包括碰撞点）上的测地线流导致其黎曼导数上的相应流：质心位形空间$C^2$和形空间$R^3$（以及E = 0时平方反比势的$S^3$和形球$S^2$）。相应的黎曼淹没明确地描述在“霍普夫”坐标，这是特别适合的等距。对于相等的质量受到平方反比的潜力，蒙哥马利表明，零能量的“一条裤子”JM度量的形状领域是测地线完成，并具有负高斯曲率，除了在拉格朗日点。我们扩展到证明的有界性和严格负的标量曲率无处不在$C^2, R^3$和$S^3$与碰撞点删除。截面曲率也被发现在很大程度上是负的，表明广泛的测地线不稳定性。我们获得渐近度量碰撞附近，表明标量曲率有有限的限制，并观察到测地线reformulation“regularizes”成对和三重碰撞$C^2$和它的任意质量和允许的能量。对于E=0的等质量牛顿势，我们发现$C^2$上的标量曲率是严格负的，尽管它在$R^3$上可以是任意一个符号.然而，与平方反比势不同的是，测地线在有限测地线时间内碰撞时会遇到曲率奇点。
英文摘要：
The Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the Jacobi-Maupertuis (JM) metric on configuration space. We study this geodesic reformulation of the planar three-body problem with both Newtonian and attractive inverse-square potentials. The associated JM metrics possess translation and rotation isometries in addition to scaling isometries for the inverse-square potential with zero energy E. The geodesic flow on the full configuration space $C^3$ (with collision points excluded) leads to corresponding flows on its Riemannian quotients: the center of mass configuration space $C^2$ and shape space $R^3$ (as well as $S^3$ and the shape sphere $S^2$ for the inverse-square potential when E = 0). The corresponding Riemannian submersions are described explicitly in `Hopf' coordinates which are particularly adapted to the isometries. For equal masses subject to inverse-square potentials, Montgomery shows that the zero-energy `pair of pants' JM metric on the shape sphere is geodesically complete and has negative gaussian curvature except at Lagrange points. We extend this to a proof of boundedness and strict negativity of scalar curvatures everywhere on $C^2, R^3$ & $S^3$ with collision points removed. Sectional curvatures are also found to be largely negative, indicating widespread geodesic instabilities. We obtain asymptotic metrics near collisions, show that scalar curvatures have finite limits and observe that the geodesic reformulation `regularizes' pairwise and triple collisions on $C^2$ and its quotients for arbitrary masses and allowed energies. For the Newtonian potential with equal masses and E=0, we find that the scalar curvature on $C^2$ is strictly negative though it could have either sign on $R^3$. However, unlike for the inverse-square potential, geodesics can encounter curvature singularities at collisions in finite geodesic time.
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