Perturbed collocation and symplectic RKN methods

It is known that many Runge-Kutta-Nyström methods can be derived by collocation. In this paper we prove that the onlys-stage symplectic Runge-Kutta-Nyström method that can be obtained by ordinary collocation is of order 2s and implicit. We also extend the idea of perturbed collocation to Runge-Kutta-Nyström methods and derive symplectic Runge-Kutta-Nyström methods using this technique. We have obtained symplectic implicits-stage Runge-Kutta-Nyström methods of order 2s?1 and 2s?2.