Kepler和限制性三体问题的Lyapunov与几何稳定性分析
Lyapunov vs. Geometrical Stability Analysis of the Kepler and the Restricted Three Body Problem
摘要：
在这封信中，我们表明，虽然应用标准的李雅普诺夫分析预测，完全可积开普勒运动是不稳定的，几何分析Horwitz等[1]预测观察到的稳定性。在我们看来，这为标准李雅普诺夫分析的不完整性和几何分析的强度提供了证据。此外，我们应用这种方法的三体问题中的第三个机构被限制在一个大半径的圆上，诱导绝热的第二个机构的时间相关的潜力。这导致第二个物体以一个非常有趣和复杂但周期性的轨迹移动;然而，标准的李雅普诺夫分析，以及基于与雅可比度量相关的曲率参数变化的方法，错误地预测了混沌行为。在这种情况下，几何方法也预测了正确的稳定运动。
英文摘要：
In this letter we show that although the application of standard Lyapunov analysis predicts that completely integrable Kepler motion is unstable, the geometrical analysis of Horwitz et al [1] predicts the observed stability. This seems to us to provide evidence for both the incompleteness of the standard Lyapunov analysis and the strength of the geometrical analysis. Moreover, we apply this approach to the three body problem in which the third body is restricted to move on a circle of large radius which induces an adiabatic time dependent potential on the second body. This causes the second body to move in a very interesting and intricate but periodic trajectory; however, the standard Lyapunov analysis, as well as methods based on the parametric variation of curvature associated with the Jacobi metric, incorrectly predict chaotic behavior. The geometric approach predicts the correct stable motion in this case as well.
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