THE COMPLETE CONSTRUCTION OF FOURTH ORDER SYMPLECTIC EXPLICIT R-K-N METHODS OF THREE STAGES
三级四阶显式辛R-K-N格式的完全构造

The sufficient and necessary conditions for a $p$-th order symmetric Runge-Kutta-Nystr\"om (R-K-N) method of s stages were expressed by a system of nonlinear equations. Thus the construction problem of symmetric schemes is transformed into the problem of solving the system. Under some special conditions, two solutions of the system for s=3 and p=4 were presented, so two fourth order symmetric explicit schemes of three stages were given. Based on Wu's method, we obtain all the solutions of the system for s=3 and p=4 by using Maple software and the software package wsolve in this paper, that is, we construct all the fourth order symplectic explicit R-K-N schemes of three stages. Moreover, we prove that the conditions for a fourth order symmetric explicit R-K-N method of three stages are redundant. Numerical results on the two-body problem indicate the higher precision of the new schemes compared to some existing schemes.