Systematic Improvement of Splitting Methods for the Hamilton Equations

We show how the standard (St{\"o}rmer-Verlet) splitting method for differential equations of Hamiltonian mechanics (with accuracy of order $\tau^2$ for a timestep of length $\tau$) can be improved in a systematic manner without using the composition method. We give the explicit expressions which increase the accuracy to order $\tau^8$, and demonstrate that the method work on a simple anharmonic oscillator.