Variational Integrators for the GravitationalN‐Body Problem

This paper describes a fourth-order integration algorithm for the gravitational N -body problem based on discrete Lagrangian mechanics. When used with shared time steps, 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 adaptive block-power-of-two time steps does not increase the error in symplecticity. In contrast to other high-order, symplectic, individual time step, momentum-preserving algorithms, the algorithm takes only forward time steps. 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 about 1.5 orders of magnitude better symplecticity and momentum conservation errors than standard algorithms for equivalent numbers of force evaluations and equivalent energy conservation errors.