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.