ABSTRACT
A Painleve analysis allowed the discovery of the solution for the rigidly rotating body in gravity where one of the moments of inertia is double the other two, and more recently has been widely applied to discover new integrable systems. Another technique for integrating Hamilton’s equations is the inverse scattering transform, which was first used to solve an infinite dimensional Hamiltonian system. Actually, the idea that Hamiltonian systems might have pseudo-random behaviour was already commonplace in the 19th century. Boltzmann founded his statistical mechanics on the hypothesis that for a gas every orbit visits all parts of its energy surface. Using the fact that Hamiltonian systems conserve phase space volume he proved that the orbit of almost every point on a compact energy surface comes back arbitrarily close to its initial condition (Poincare recurrence). For Hamiltonian systems of two degrees of freedom, the invariant tori are two-dimensional and divide the three-dimensional energy surface.
