  The New Horizons flyby of the cold classical Kuiper Belt object MU69 showed
it to be a contact binary. The existence of other contact binaries in the
1-10km range raises the question of how common these bodies are and how they
evolved into contact. Here we consider that the lobes of MU69 formed as a
binary in the Solar nebula, and calculate its orbital evolution in the presence
of gas. We find that the sub-Keplerian wind of the disk brings the drag
timescales for 10km bodies under 1 Myr for quadratic-velocity drag. In the
Kuiper belt, however, the drag is linear with velocity and the effect of the
wind cancels out as the angular momentum gained in half an orbit is lost in the
other half: the drag timescales for 10 km bodies remain over 10 Myr. In this
situation we find that a combination of drag and Kozai-Lidov oscillations is a
promising channel for collapse. We analytically solve the hierarchical
three-body problem with drag and implement it into a Kozai plus tidal friction
model. The permanent quadrupoles of the lobes make the Kozai oscillations
stochastic, and as drag shrinks the semimajor axis it more easily allows the
fluctuations to bring the system into contact. Evolution to contact happens
very rapidly (within $10^4$ yr) in the pure, double-average quadrupole, Kozai
region between $\approx 85-95^\circ$, and within 3 Myr in the drag-assisted
region beyond it. The synergy between $J_2$ and gas drag widens the window of
contact to 80-100$^\circ$ initial inclination, over a larger range of semimajor
axes. As such, the model predicts an initial contact binary fraction of about
10\% for the cold classicals in the Kuiper belt. The speed at contact deviates
from the escape velocity only because of the oblateness. For MU69, the contact
velocity should be 3.3-4.2 m/s, in line with the observational evidence from
the lack of deformation features and estimate of the tensile strength.
