Symplectic integration for the collisional gravitational $N$-body problem

We present a new symplectic integrator designed for collisional gravitational $N$-body problems which makes use of Kepler solvers. The integrator is also reversible and conserves 9 integrals of motion of the $N$-body problem to machine precision. The integrator is second order, but the order can easily be increased by the method of \citeauthor{yos90}. We use fixed time step in all tests studied in this paper to ensure preservation of symplecticity. We study small $N$ collisional problems and perform comparisons with typically used integrators. In particular, we find comparable or better performance when compared to the 4th order Hermite method and much better performance than adaptive time step symplectic integrators introduced previously. We find better performance compared to SAKURA, a non-symplectic, non-time-reversible integrator based on a different two-body decomposition of the $N$-body problem. The integrator is a promising tool in collisional gravitational dynamics.