Variational Integrators for the Gravitational N-Body Problem

This paper describes a novel fourth-order integration algorithm for the gravitational N-body problem based on discrete Lagrangian mechanics. When used with shared timesteps, the algorithm is momentum conserving and symplectic. We generalize the algorithm to handle individual time steps; this introduces fifth-order errors in angular momentum conservation and symplecticity. We show that using block power of two timesteps does not increase the error in symplecticity. In contrast to other high-order, symplectic, momentum-preserving algorithms in widespread astrophysical use, the algorithm takes only forward timesteps. We compare a code integrating an N-body system using the algorithm with a direct-summation force calculation to standard stellar cluster simulation codes. We find that our algorithm has significantly better symplecticity and momentum conservation errors than standard algorithms for equivalent numbers of potential evaluations and equivalent energy conservation errors