Lyapunov vs. Geometrical Stability Analysis of the Kepler and the Restricted Three Body Problem

TL;DR

This paper compares Lyapunov and geometrical stability analyses for celestial mechanics, showing the geometrical approach better predicts stability in Kepler and restricted three-body problems.

Contribution

It demonstrates the limitations of Lyapunov analysis and highlights the effectiveness of geometrical methods in predicting stability in celestial systems.

Findings

Lyapunov analysis predicts instability where geometrical analysis predicts stability.

Geometrical analysis accurately predicts stability in Kepler and restricted three-body problems.

Standard Lyapunov methods incorrectly predict chaos in certain celestial motions.

Abstract

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.