  The Milky Way (MW) and M31 both harbor massive satellite galaxies, the Large
Magellanic Cloud (LMC) and M33, which may comprise up to 10 per cent of their
host's total mass. Massive satellites can change the orbital barycentre of the
host-satellite system by tens of kiloparsecs and are cosmologically expected to
harbor dwarf satellite galaxies of their own. Assessing the impact of these
effects depends crucially on the orbital histories of the LMC and M33. Here, we
revisit the dynamics of the MW-LMC system and present the first detailed
analysis of the M31-M33 system utilizing high precision proper motions and
statistics from the dark matter-only Illustris cosmological simulation. With
the latest Hubble Space Telescope proper motion measurements of M31, we
reliably constrain M33's interaction history with its host. In particular, like
the LMC, M33 is either on its first passage (t_{inf} < 2 Gyr ago) or if M31 is
massive (>=2x10^12 Msun), it is on a long period orbit of about 6 Gyr.
Cosmological analogs of the LMC and M33 identified in Illustris support this
picture and provide further insight about their host masses. We conclude that,
cosmologically, massive satellites like the LMC and M33 are likely completing
their first orbits about their hosts. We also find that the orbital energies of
such analogs prefer a MW halo mass ~1.5x10^12 Msun and an M31 halo mass
>=1.5x10^12 Msun. Despite conventional wisdom, we conclude it is highly
improbable that M33 made a close (< 100 kpc) approach to M31 recently (
t_{peri} < 3 Gyr ago). Such orbits are rare (< 1 per cent) within the 4$\sigma$
error space allowed by observations. This conclusion cannot be explained by
perturbative effects through four body encounters between the MW, M31, M33, and
the LMC. This surprising result implies that we must search for a new
explanation for M33's strongly warped gas and stellar discs.
