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We develop continuous-stage Runge-Kutta-Nystr\"{o}m (csRKN) methods forsolving second order ordinary differential equations (ODEs) in this paper. Thesecond order ODEs are commonly encountered in various fields and some of themcan be reduced to the first order ODEs with the form of separable Hamiltoniansystems. The symplecticity-preserving numerical algorithm is of interest forsolving such special systems. We present a sufficient condition for a csRKNmethod to be symplecticity-preserving, and by using Legendre polynomialexpansion we show a simple way to construct such symplectic RKN type method.

Symplecticity-preserving continuous-stage Runge-Kutta-Nystr\"{o}m methods