On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry

This paper presents a detailed comparison of two implicit time integration schemes for a simple non-linear Hamiltonian system with symmetry: the motion of a particle in a central force field. The goal is to establish analytical and numerical results pertaining to the stability properties of the implicit mid-point rule (the proto-typical implicit symplectic method) and a particular energy-momentum conserving scheme, and to compare the two schemes with respect to accuracy. While all results presented herein are within the context of a simple model problem, the problem was constructed so as to exhibit key features typical of more complex systems with symmetry such as those arising in non-linear solid mechanics: namely, the presence of large (and relatively slow) overall motions together with high-frequency internal motions.