http://arxiv.org/abs/1904.03364
Symplectic integrators are a foundation to the study of dynamical $N$-body phenomena, at scales ranging from from planetary to cosmological. These integrators preserve the Poincar\'{e} invariants of Hamiltonian dynamics. The $N$-body Hamiltonian has another, perhaps overlooked, symmetry: it is perfectly smooth for particle separations greater than $0$. Many popular symplectic integrators, such as hybrid methods or block adaptive stepping methods, do not respect this symmetry and it is perhaps unclear whether they should. We investigate the importance of this symmetry by considering hybrid integrators, whose smoothness can be tuned easily. Hybrid methods are perfectly smooth, except at a finite number of phase space points. We study chaotic planetary orbits in a test considered by Wisdom. We find that increasing smoothness, at negligible extra computational cost in particular tests, improves the Jacobi constant error of the orbits by about $5$ orders of magnitude in long-term simulations. The results from this work suggest that smoothness of the $N$-body problem is a symmetry worth preserving in simulations.
D. Hernandez
Tue, 9 Apr 19
4/105
Comments: 8 pages, 7 figures, submitted to MNRAS, comments welcome
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