  We study dynamical systems which admit action-angle variables at leading
order which are subject to nearly resonant perturbations. If the frequencies
characterizing the unperturbed system are not in resonance, the long-term
dynamical evolution may be integrated by orbit-averaging over the
high-frequency angles, thereby evolving the orbit-averaged effect of the
perturbations. It is well known that such integrators may be constructed via a
canonical transformation, which eliminates the high frequency variables from
the orbit-averaged quantities. An example of this algorithm in celestial
mechanics is the von Zeipel transformation. However if the perturbations are
inside or close to a resonance, i.e. the frequencies of the unperturbed system
are commensurate, these canonical transformations are subject to divergences.
We introduce a canonical transformation which eliminates the high frequency
phase variables in the Hamiltonian without encountering divergences. This leads
to a well-behaved symplectic integrator. We demonstrate the algorithm through
two examples: a resonantly perturbed harmonic oscillator and the gravitational
three-body problem in mean motion resonance.
