Footprints of the Sagittarius dwarf galaxy in the $Gaia$ data set
  We analyse an N-body simulation of the interaction of the Milky Way (MW) with
a Sagittarius-like dSph (Sgr), looking for signatures which may be attributed
to its orbital history in the phase space volume around the Sun in light of
$Gaia$ DR2 discoveries. The repeated impacts of Sgr excite coupled vertical and
radial oscillations in the disc which qualitatively, and to a large degree
quantitatively are able to reproduce many features in the 6D $Gaia$ DR2
samples, from the median $V_{R},V_{\phi}, V_{z} $ velocity maps to the local
$\delta\rho(v_{z},z)$ phase-space spiral which is a manifestation of the global
disc response to coupled oscillations within a given volume. The patterns in
the large-scale velocity field are well described by tightly wound spirals and
vertical corrugations excited from Sgr's impacts. We show that the last
pericentric passage of Sgr resets the formation of the local present-day
$\delta\rho(v_z,z)$ spiral and situate its formation around 500-800 Myr. As
expected $\delta\rho(vz,z)$ grows in size and decreases in woundedness as a
function of radius in both the $Gaia$ DR2 data and simulations. This is the
first N-body model able to explain so many of the features in the data on
different scales. We demonstrate how to use the full extent of the Galactic
disc to date perturbations dating from Myr to Gyr, probe the underlying
potential and constrain the mass-loss history of Sgr. $\delta\rho(vz,z)$ looks
the same in all stellar populations age bins down to the youngest ages which
rules out a bar buckling origin.
