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