Mixed-variable symplectic integrators provide a fast, moderately accurate way to study the long-term evolution of a wide variety of N-body systems (Wisdom & Holman 1991). They are especially suited to planetary and satellite systems, in which a central body contains most of the mass. However, in their original form, they become inaccurate whenever two bodies approach one another closely. Here, I will show how to overcome this difficulty using a hybrid integrator that combines symplectic and conventional algorithms.

A symplectic integrator works by splitting the Hamiltonian, H, for an N-body system, into two or more parts H = H0 + H1 + …, where є i = Hi/H0 ≪ 1 for i = 1, 2 …. An integration step consists of several substeps, each of which advances the system due to the effect of one part of the Hamiltonian only. The error incurred over the whole step is ∼ є τn1, where τ is the timestep, n is the order of the integrator, and є is the largest of єi.