Should $N$-body integrators be symplectic everywhere in phase space?

Abstract

Symplectic integrators are the preferred method of solving conservative $N$-body problems in cosmological, stellar cluster, and planetary system simulations because of their superior error properties and ability to compute orbital stability. Newtonian gravity is scale free, and there is no preferred time or length scale: this is at odds with construction of traditional symplectic integrators, in which there is an explicit timescale in the time-step. Additional timescales have been incorporated into symplectic integration using various techniques, such as hybrid methods and potential decompositions in planetary astrophysics, integrator sub-cycling in cosmology, and block time-stepping in stellar astrophysics, at the cost of breaking or potentially breaking symplecticity at a few points in phase space. The justification provided, if any, for this procedure is that these trouble points where the symplectic structure is broken should be rarely or never encountered in practice. We consider the case of hybrid integrators, which are used ubiquitously in astrophysics and other fields, to show that symplecticity breaks at a few points are sufficient to destroy beneficial properties of symplectic integrators, which is at odds with some statements in the literature. We show how to solve this problem in the case of hybrid integrators by requiring Lipschitz continuity of the equations of motion. For other techniques, like time step subdivision, {consequences to this this problem are not explored here}, and the fact that symplectic structure is broken should be taken into account by $N$-body simulators, who may find an alternative non-symplectic integrator performs similarly.

arXiv comment8 pages, 9 figures. Matches accepted MNRAS version

References in corpus (10)

• The cosmological simulation code GADGET-2
• Hybrid Symplectic Integrators for Planetary Dynamics
• A new approach to simulating collisionless dark matter fluids
• Secular dynamics of hierarchical multiple systems composed of nested binaries, with an arbitrary number of bodies and arbitrary hierarchical structure. First applications to multiplanet and multistar systems
• The action spectrum and C^0 symplectic topology
• A study of symplectic integrators for planetary system problems: error analysis and comparisons
• Variational Integrators for the Gravitational N-Body Problem
• Symplectic integration for the collisional gravitational $N$-body problem
• Fast and reliable symplectic integration for planetary system $N$-body problems
• Time-symmetric integration in astrophysics

Cited by in corpus (10)

• REBOUNDx: A Library for Adding Conservative and Dissipative Forces to Otherwise Symplectic N-body Integrations
• Hybrid Symplectic Integrators for Planetary Dynamics
• EXP: N-body integration using basis function expansions
• Are long-term $N$-body simulations reliable?
• Stepsize errors in the $N$-body problem: discerning Mercury's true possible long-term orbits
• Switching integrators reversibly in the astrophysical $N$-body problem
• Physical properties of terrestrial planets and water delivery in the habitable zone using N-body simulations with fragmentation
• EnckeHH: an integrator for gravitational dynamics with a dominant mass that achieves optimal error behaviour
• Improving the accuracy of simulated chaotic $N$-body orbits using smoothness
• KRIOS: A new basis-expansion $N$-body code for collisional stellar dynamics