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It is common in classical mechanics to encounter systems whose Hamiltonian$H$ is the sum of an often exactly integrable Hamiltonian $H_0$ and a smallperturbation $\epsilon H_1$ with $\epsilon\ll1$. Such near-integrability can beexploited to construct particularly accurate operator splitting methods tosolve the equations of motion of $H$. However, in many cases, for example inproblems related to planetary motion, it is computationally expensive to obtainthe exact solution to $H_0$.  In this paper we present a new family of embedded operator splitting (EOS)methods which do not use the exact solution to $H_0$, but rather approximate itwith yet another, embedded operator splitting method. Our new methods have allthe desirable properties of classical methods which solve $H_0$ directly. Butin addition they are very easy to implement and in some cases faster. Whenapplied to the problem of planetary motion, our EOS methods have error scalingsidentical to that of the often used Wisdom-Holman method but do not require aKepler solver, nor any coordinate transformations, or the allocation of memory.The only two problem specific functions that need to be implemented are thestraight-forward kick and drift steps typically used in the standard secondorder leap-frog method.

Embedded operator splitting methods for perturbed systems