Variable Timestep Integrators for Long-Term Orbital Integrations

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Symplectic integration algorithms have become popular in recent years in long-term orbital integrations because these algorithms enforce certain conservation laws that are intrinsic to Hamiltonian systems. For problems with large variations in timescale, it is desirable to use a variable timestep. However, naively varying the timestep destroys the desirable properties of symplectic integrators. We discuss briefly the idea that choosing the timestep in a time symmetric manner can improve the performance of variable timestep integrators. Then we present a symplectic integrator which is based on decomposing the force into components and applying the component forces with different timesteps. This multiple timescale symplectic integrator has all the desirable properties of the constant timestep symplectic integrators.