  We explore the structural and kinematic properties of the outskirts of the
Large Magellanic Cloud (LMC) using data from the Magellanic Edges Survey
(MagES) and Gaia EDR3. Even at large galactocentric radii
($8^\circ<R<11^\circ$), we find the north-eastern LMC disk is relatively
unperturbed: its kinematics are consistent with a disk of inclination
~$36.5^\circ$ and line-of-nodes position angle ~$145^\circ$ east of north. In
contrast, fields at similar radii in the southern and western disk are
significantly perturbed from equilibrium, with non-zero radial and vertical
velocities, and distances significantly in front of the disk plane implied by
our north-eastern fields. We compare our observations to simple dynamical
models of the Magellanic/Milky Way system which describe the LMC as a
collection of tracer particles within a rigid potential, and the Small
Magellanic Cloud (SMC) as a rigid Hernquist potential. A possible SMC crossing
of the LMC disk plane ~400 Myr ago, in combination with the LMC's infall to the
Milky Way potential, can qualitatively explain many of the perturbations in the
outer disk. Additionally, we find the claw-like and arm-like structures south
of the LMC have similar metallicities to the outer LMC disk ([Fe/H]~-1), and
are likely comprised of perturbed LMC disk material. The claw-like substructure
is particularly disturbed, with out-of-plane velocities >60 km s$^{-1}$ and
apparent counter-rotation relative to the LMC's disk motion. More detailed
N-body models are necessary to elucidate the origin of these southern features,
potentially requiring repeated interactions with the SMC prior to ~1 Gyr ago.
