Symplectic integration of Hamiltonian systems

The authors 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 Poincare invariants. The algorithms have been tested 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.

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