An Explicit Runge-Kutta-Nyström Method is Canonical If and Only If Its Adjoint is Explicit

Canonical Runge-Kutta-Nyström methods for Hamiltonian dynamical systems are considered. These systems arise in various areas in the physical sciences. Canonical methods preserve certain properties of the system. In this paper, it is shown that an explicit Runge-Kutta-Nyström method is canonical if and only its adjoint is explicit. The adjoint of a method is obtained by time reversal. One application for this result and other results in this paper is that if an explicit canonical Runge-Kutta-Nyström method of odd order is concatenated with its adjoint the result is an explicit canonical method of one order higher.