Symplectic Integration of Hamiltonian Systems

Abstract

We survey past work and present new algorithms to numerically integrate the trajectories of Hamiltonian dynamical systems. These algorithms exactly preserve the symplectic 2-form, i.e. they preserve all the Poincaré invariants. The algorithms have been tasted on a variety of examples and results are presented for the Fermi-Pasta-Ulam nonlinear string, the Henon-Heiles system, a four-vortex problem, and the geodesic flow on a manifold of constant negative curvature. In all cases the algorithms possess long-time stability and preserve global geometrical structures in phase space.