ORDER CONDITIONS FOR CANONICAL RUNGE-KUTTA-NYSTROM METHODS

We are concerned with Runge-Kutta-Nystrom methods for the integration of second order systems of the special form d2y/dt2 = f(y). If the function f is the gradient of a scalar field, then the system is Hamiltonian and it may be advantageous to integrate it by a so-called canonical Runge-Kutta-Nystrom formula. We show that the equations that must be imposed on the coefficients of the method to ensure canonicity are simplifying assumptions that lower the number of.independent order conditions. We count the number of order conditions, both for general and for canonical Runge-Kutta-Nystrom formulae.