Construction of higher order symplectic integrators

For Hamiltonian systems of the form H = T( p)+ V( q) a method is shown to construct explicit and time reversible symplectic integrators of higher order. For any even order there exists at least one symplectic integrator with exact coefficients. The simplest one is the 4th order integrator which agrees with one found by Forest and by Neri. For 6th and 8th orders, symplectic integrators with fewer steps are obtained, for which the coefficients are given by solving a set of simultaneous algebraic equations numerically.