Symplectic integrators and their application to dynamical astronomy

AbstractSymplectic 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 c as a factor, the step size of an nth order symplectic integrator can be lengthened by a factor ε−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.