HIGH-ORDER SYMPLECTIC RUNGE-KUTTA-NYSTROM METHODS

A numerical method for ordinary differential equations is called sympl ectic if, when applied to Hamiltonian problems, it preserves the sympl ectic structure in phase space, thus reproducing the main qualitative property of solutions of Hamiltonian systems. The authors construct an d test symplectic, explicit Runge-Kutta-Nystrom (RKN) methods of order 8. The outcome of the investigation is that existing high-order, symp lectic RKN formulae require so many evaluations per step that they are much less efficient than conventional eighth-order nonsymplectic, var iable-step-size integrators even for low accuracy. However, symplectic integration is of use in the study of qualitative features of the sys tems being integrated.