Simpson's quadrature for a nonlinear variational symplectic scheme

We propose a variational symplectic numerical method for the time integration of dynamical systems issued from the least action principle. We assume a quadratic internal interpolation of the state between two time steps and we approximate the action in one time step by the Simpson's quadrature formula. The resulting scheme is nonlinear and symplectic. First numerical experiments concern a nonlinear pendulum and we have observed experimentally very good convergence properties.