  We report the first 3-D kinematical measurements of 88 stars in the direction
of several recently discovered substructures in the southern periphery of the
Large Magellanic Cloud (LMC) using a combination of Gaia proper motions and
radial velocities from the APOGEE-2 survey. More specifically, we explore stars
lie in assorted APOGEE-2 pointings in a region of the LMC periphery where
various overdensities of stars have previously been identified in maps of stars
from Gaia and DECam. By using a model of the LMC disk rotation, we find that a
sizeable fraction of the APOGEE-2 stars have extreme space velocities that are
distinct from, and not a simple extension of, the LMC disk. Using N-body
hydrodynamical simulations of the past dynamical evolution and interaction of
the LMC and Small Magellanic Cloud (SMC), we explore whether the extreme
velocity stars may be accounted for as tidal debris created in the course of
that interaction. We conclude that the combination of LMC and SMC debris
produced from their interaction is a promising explanation, although we cannot
rule out other possible origins, and that these new data should be used to
constrain future simulations of the LMC-SMC interaction. We also conclude that
many of the stars in the southern periphery of the LMC lie out of the LMC plane
by several kpc. Given that the metallicity of these stars suggest they are
likely of Magellanic origin, our results suggest that a wider exploration of
the past interaction history of the Magellanic Clouds is needed.
