  Motivated by recent studies suggesting that the Large Magellanic Cloud (LMC)
could be significantly more massive than previously thought, we explore whether
the approximation of an inertial Galactocentric reference frame is still valid
in the presence of such a massive LMC. We find that previous estimates of the
LMC's orbital period and apocentric distance derived assuming a fixed Milky Way
are significantly shortened for models where the Milky Way is allowed to move
freely in response to the gravitational pull of the LMC. Holding other
parameters fixed, the fraction of models favoring first infall is reduced. Due
to this interaction, the Milky Way center of mass within the inner 50 kpc can
be significantly displaced in phase-space in a very short period of time that
ranges from 0.3 to 0.5 Gyr by as much as 30 kpc and 75 km/s. Furthermore, we
show that the gravitational pull of the LMC and response of the Milky Way are
likely to significantly affect the orbit and phase space distribution of tidal
debris from the Sagittarius dwarf galaxy (Sgr). Such effects are larger than
previous estimates based on the torque of the LMC alone. As a result, Sgr
deposits debris in regions of the sky that are not aligned with the present-day
Sgr orbital plane. In addition, we find that properly accounting for the
movement of the Milky Way around its common center of mass with the LMC
significantly modifies the angular distance between apocenters and tilts its
orbital pole, alleviating tensions between previous models and observations.
While these models are preliminary in nature, they highlight the central
importance of accounting for the mutual gravitational interaction between the
MW and LMC when modeling the kinematics of objects in the Milky Way and Local
Group.
