High order symplectic integrators for perturbed Hamiltonian systems

Abstract. We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H = A + εB. We give a constructive proof that for all integer p, there exists an integrator with positive steps with a remainder of order O(τε + τ ε), where τ is the stepsize of the integrator. The analytical expressions of the leading terms of the remainders are given at all orders. In many cases, a corrector step can be performed such that the remainder becomes O(τε + τ ε). The performances of these integrators are compared for the simple pendulum and the planetary 3-Body problem of Sun-Jupiter-Saturn.