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Abstract. In this paper various aspect of symplectic integrators are reviewed. Symplectic integrators are numerical integration methods for Hamiltonian systems which are designed to conserve the symplectic structure exactly as the original flow. There are explicit symplectic schemes for systems of the form H = T(p) + V(q). and implicit schemes for general Hamiltonian systems. As a general property. symplectic integrators conserve the energy quite well and therefore an artificial damping (excitation) caused by the accumulation of the local truncation error cannot occur. Symplectic integrators have been applied to the Kepler problem. the motion of minor bodies in the solar system and the long-term evolution of outer planets.

Symplectic integrators have many merits compared with traditional integrators: -the numerical solutions have a property of area preserving, -the discretization error in the energy integral does not have a secular term, which means that the accumulated truncation errors in angle variables increase linearly with the time instead of quadratic growth, -the symplectic integrators can integrate an orbit with high eccentricity without change of step-size. The symplectic integrators discussed in this paper have the following merits in addition to the previous merits: -the angular momentum vector of the nbody problem is exactly conserved, -the numerical solution has a property of time reversibility, -the truncation errors, especially the secular error in the angle variables, can easily be estimated by an usual perturbation method, -when a Hamiltonian has a disturbed part with a small parameter e as a factor, the step size of an nth order symplectic integrator can be lengthened by a factor e -1/n with use of two canonical sets of variables, -the number of evaluation of the force function by the 4th order symplectic integrator is smaller than the classical Runge-Kutta integrator method of the same order. The symplectic integrators are well suited to integrate a Hamiltonian system over a very long time span.

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