Symplecticness, stability, and asymptotic properties of Runge-Kutta, partitioned Runge Kutta, and Runge-Kutta-Nystrom methods applied to the simple Hamiltonian system p = -vq, q = κp are studied. Some new results in connection with P-stability are pre sented. The main part is focused on backward error analysis. The numerical solution produced by a symplectic method with an appropriate stepsize is the exact solution of a perturbed Hamiltonian system at discrete points. This system is studied in detail and new results are derived. Numerical examples are presented.