first and last steps can be combined if the system is evolved for multiple timesteps.

Note that the evolution of $\operatorname{\mathcal{H}}_{\rm Kepler}$ and $\operatorname{\mathcal{H}}_0$ is most easily accomplished in Jacobi coordinates. The interaction Hamiltonian $\operatorname{\mathcal{H}}_{\rm Interaction}$, however, contains terms that depend on both the Cartesian and Jacobi coordinates. The simplest way to calculate these terms is to convert to Cartesian coordinates, evaluate the $\mathbf{r}_i-\mathbf{r}_j$ term, convert the accelerations back to Jacobi accelerations, and calculate the remaining terms.

Symplectic Correctors {#sec:correctors}
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The operator splitting method used in the symplectic integrator discussed above effectively adds high frequency terms to the Hamiltonian. An argument often used in favour of symplectic integrators is that, although these high-frequency terms alter the Hamiltonian, they do not change the long term evolution as they average out. However, they do lead to relatively large short term oscillations, for example in the energy error.

The idea of a symplectic corrector, first used by [@TittemoreWisdom1989] and fully developed by [@Wisdom1996], is to remove some of these high frequency terms using perturbation theory. The basic procedure is as follows. Before the start of an integration, we convert from real coordinates to so-called mapping coordinates. Then we perform the integration using 