The accuracy of symplectic integrators

Introduction The symplectic integration of Hamiltonian dynamical systems is by now an established technique. Ruth [1] has developed explicit methods for separable systems; his approach was extended to fourth order by Candy and Rozmus [2]. Channel and Scovel [3] and Feng et al. [4] have derived methods based on the Taylor series expansion of the time map of a general Hamiltonian. Feng [5] contains a survey of the Chinese program and an important generalisation which includes many known methods as special cases. In [6] Feng provides a discussion of the philosophy and history of symplectic integration. For a summary of explicit symplectic integrators for separable Hamiltonians, see (2.2) and Table II. Standard integrators do not generally preserve the Poincar'e integral invariants of a Hamiltonian flow and cannot hope to capture the long-time dynamics of the system. Typically their numerical diffusion causes orbits to be attracted to elliptic orbits, or, coupled with forcing, cre