Order properties of symplectic runge-kutta-nyström methods

In this paper, some characteristics and order properties of symplectic Runge-Kuttar-Nyström (RKN) methods are given. By using a transformation technique, a family of high-order implicit symplectic RKN methods of order 2s -1 or order 2s is constructed, and some available symplectic RKN methods including singly-implicit, multiply-implicit, and diagonally-implicit symplectic RKN methods are investigated. As examples, two-stage and three-stage symplectic RKN methods are derived in detail.