Forward symplectic integrators for solving gravitational few-body problems

We introduce a class of fourth order symplectic algorithms that are ideal for doing long time integration of gravitational few-body problems. These algorithms have only positive time steps, but require computing the force gradient in addition to the force. We demonstrate the efficiency of these Forward Symplectic Integrators by solving the circular restricted three-body problem in the space-fixed frame where the force on the third body is explicitly time-dependent. These algorithms can achieve accuracy of Runge-Kutta, conventional negative time step symplectic and corrector symplectic algorithms at step sizes five to ten times as large.