A Class of Symplectic Integrators with Adaptive Time Step for Separable Hamiltonian Systems

Symplectic integration algorithms are well suited for long-term integrations of Hamiltonian systems, because they preserve the geometric structure of the Hamiltonian flow. However, this desirable property is generally lost when adaptive time step control is added to a symplectic integrator. We describe an adaptive time step, symplectic integrator that can be used if the Hamiltonian is the sum of kinetic and potential energy components and the required time step depends only on the potential energy (e.g., test-particle integrations in fixed potentials). In particular, we describe an explicit, reversible, symplectic, leapfrog integrator for a test particle in a near-Keplerian potential; this integrator has a time step proportional to distance from the attracting mass and has the remarkable property of integrating orbits in an inverse-square force field with only ``along-track'' errors; i.e., the phase-space shape of a Keplerian orbit is reproduced exactly, but the orbital period is in error by O(N^-2), where N is the number of steps per period.