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On the basis of the previous work by Tang \& Zhang (Appl. Math. Comput. 323,2018, p. 204--219), in this paper we present a more effective way to constructhigh-order symplectic integrators for solving second order Hamiltonianequations. Instead of analyzing order conditions step by step as shown in theprevious work, the new technique of this paper is using Legendre expansions todeal with the simplifying assumptions for order conditions. With the newtechnique, high-order symplectic integrators can be conveniently devised bytruncating an orthogonal series.

High order symplectic integrators based on continuous-stage Runge-Kutta Nystrom methods