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