A new class of symplectic integration schemes based on generating functions

We present a new family of one-step symplectic integration schemes for Hamiltonian systems of the general form ẏ = J∇H(y) . Such a class of methods contains as particular cases the methods of Miesbach and Pesch [13], and also the family of symplectic Runge-Kutta methods. As in the case of the methods introduced in [13], the new integration methods are constructed by defining a generating function, which automatically determines a symplectic map. The resulting methods are implicit, and require the evaluation of the gradient of the Hamiltonian function as well as the Hessian times a vector.