On the Error Reduction of a Simple Symplectic Integrator

An analysis of truncation errors of the simplest symplectic integrator is revisited. We performed an analytical error estimate of this integrator within the framework of a canonical perturbation method, taking the two-body system as an example. Our study was motivated by the desire to remove any confusion in the derivation of this issue in one standard literature along this line of research. We repeated the derivation, and examined how a secular numerical error arises in symplectic integration. We showed that it is possible to substantially eliminate a secular numerical error by choosing appropriate initial conditions, as previous studies have shown. On the other hand, depending on boundary parameter values, such as the eccentricity in the two-body system, we could see that it is sometimes impossible to reduce the secular numerical error by changing the starting conditions. We also numerically demonstrated this reduction, or lack of reduction of numerical errors by taking a restricted three-body system as an example. We confirmed that the so-called "iterative start" of symplectic integrators works in certain systems.