A note on the algorithm of symplectic integrators

For a Hamiltonian that can be separated into N+1(N\geq 2) integrable parts, four algorithms can be built for a symplectic integrator. This research compares these algorithms for the first and second order integrators. We found that they have similar local truncation errors represented by error Hamiltonian but rather different numerical stability. When the computation of the main part of the Hamiltonian, H0, is not expensive, we recommend to use S* type algorithm, which cuts the calculation of the H0 system into several small time steps as Malhotra(1991) did. As to the order of the N+1 parts in one step calculation, we found that from the large to small would get a slower error accumulation.