# 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.