Studies in Numerical Nonlinear Instability III: Augmented Hamiltonian Systems

Leap-frog (explicit mid-point) discretizations of Hamiltonian systems are considered. It is proved that the discrete evolution preserves the symplectic structure of the phase space. The Kolmogorov–Arnold–Moser (KAM) theory is applied to guarantee the boundedness of the computed points under suitable restrictions of the time-step. The general ideas introduced are employed to provide an extensive analysis of the leap-frog discretization of the complex equation izₜ + | z |² z = 0, which describes the nonlinear selfinteraction of a Fourier mode. The paper emphasizes the relations between Numerical Analysis and Classical Mechanics/ Dynamical Systems.