# Improving the accuracy of simulated chaotic $N$-body orbits using smoothness

## TL;DR

This paper demonstrates that increasing the smoothness of symplectic integrators in N-body simulations significantly improves long-term accuracy, highlighting the importance of Hamiltonian smoothness in dynamical studies.

## Contribution

It introduces the idea that preserving Hamiltonian smoothness in hybrid integrators enhances simulation accuracy without extra computational cost.

## Key findings

- Smooth hybrid integrators reduce Jacobi constant error by about 5 orders of magnitude.
- Increasing smoothness improves long-term stability of chaotic orbits.
- Smoothness of the N-body Hamiltonian is a crucial property for accurate simulations.

## Abstract

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 smooth, or, in other words, it has infinite differentiability class order (DCO) for particle separations greater than $0$. Popular symplectic integrators, such as hybrid methods or block adaptive stepping methods do not come from smooth Hamiltonians and it is perhaps unclear whether they should. We investigate the importance of this symmetry by considering hybrid integrators, whose DCO can be tuned easily. Hybrid methods are 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 Hamiltonian is a property worth preserving in simulations.