Subtle is the lord: On the difference between Newtonian (Lyapunov) stability analysis and geometrical stability analysis of gravitational orbits

In this essay we discuss the geometrical embedding method (GEM) for the analysis of the stability of Hamiltonian systems using geometrical techniques familiar from general relativity. This method has proven to be very effective. In particular, we show that although the application of standard Lyapunov analysis predicts that completely integrable Kepler motion is unstable, this geometrical analysis predicts the observed stability. 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 intricate but periodic trajectory. The geometric approach predicts the correct stable motion in this case as well.