ABSTRACT
This chapter discusses 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 sometimes contrary to indications of the Lyapunov method. In 1898, when Hadamard Billiard studied geodesics on surfaces of negative curvature, he wrote a seminal paper “On the Billard on a Surface with Negative Curvature”. In the traditional geometrical approach describing Hamiltonian chaos with tools from Riemannian geometry the natural motions of Hamiltonian systems are viewed as geodesics of the configuration space manifold equipped with a metric. Yurtsever identified the geometric sources of the chaotic dynamics by reducing the problem to that of geodesic motion on a negatively curved surface. The Hamilton equations of the original potential model are in a geometrical approach contained in the geodesics equations through an inverse map in the tangent space in terms of a geometric embedding.
