Geometry of local instability in hamiltonian dynamics

In this chapter, we discuss a geometrical embedding method for the analysis of the stability of time-dependent Hamiltonian systems using geometrical techniques familiar from general relativity. This method has proven to be very effective in numerous examples, predicting correctly the stability/instability of motions, sometimes contrary to indications of the Lyapunov method. For example, although the application of local Lyapunov analysis predicts the completely integrable Kepler motion to be unstable, this geometrical analysis predicts the observed stability The general theory of the structure and application of this method for time-independent and time-dependent potential problems is given in this chapter, with criteria for its applicabilityin these cases. As time-independent examples, the perturbed 2D coupled harmonic oscillator and a fifth-order expansion Toda lattice are studied in more detail and the controlling effect is demonstrated. Two time-dependent examples are presented: the restricted three-body problem and the two-dimensional Duffing oscillator with a time-dependent coefficient. The first represents an important class of geophysical and astrophysical problems, and the second, the result of a perturbed bistable system with analogs in electric circuit theory In both cases the local Lyapunov analysis fails to predict the correct limiting behavior.