Hamiltonian Approximants for Symplectic Integrators

William I. Newman and Jeremy Schiff

Symplectic integrators do not, in general, reproduce all the features of the dynamics of the Hamiltonian systems which they approximate. For example, energy conservation is lost, and global features such as separatrices can be destroyed. We study these effects for a Hamiltonian system with a single degree of freedom and the simplest possible symplectic integrator. We look at a sequence of Hamiltonian systems of higher and higher dimension, that interpolate between the original Hamiltonian system and the symplectic integrator. In these intermediate Hamiltonian systems we can make concrete statements about energy conservation and separatrix splitting. The qualitative dynamics of the symplectic integrator seems to be inherited from these intermediate systems, and in some cases we can even deduce quantitative results for the symplectic integrator from those of the intermediate systems.