High order symplectic integrators for perturbed Hamiltonian systems

A family of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H = A + epsilonB was given in (McLachlan, 1995). We give here a constructive proof that for all integer p, such integrator exists, with only positive steps, and with a remainder of order O(tau (p)epsilon + tau (2)epsilon (2) ), where tau is the stepsize of the integrator. Moreover, we compute the analytical expressions of the leading terms of the remainders at all orders. We show also that for a large class of systems, a corrector step can be performed such that the remainder becomes O(tau (p)epsilon + tau (4)epsilon (2)). The performances of these integrators are compared for the simple pendulum and the planetary three-body problem of Sun-Jupiter-Saturn.