  We discuss systematic extensions of the standard (St{\"o}rmer-Verlet)
splitting method for differential equations of Hamiltonian mechanics, with
relative accuracy of order $\tau^2$ for a timestep of length $\tau$, to higher
orders in $\tau$. We present some splitting schemes, with all intermediate
timesteps real and positive, which increase the relative accuracy to order
$\tau^{N}$ (for N=4, 6, and 8) for a large class of Hamiltonian systems.
