  The goal of this paper is to show existence of short-time classical solutions
to the so called Master Equation of \emph{first order} Mean Field Games, which
can be thought of as the limit of the corresponding master equation of a
stochastic mean field game as the individual noises approach zero. Despite
being the equation of an idealistic model, its study is justified as a way of
understanding mean field games in which the individual players'~randomness is
negligible; in this sense it can be compared to the study of ideal fluids
\cite{gangboberkeleynotes}. We restrict ourselves to potential mean field games
but do not impose any monotonicity conditions on the running and initial costs,
and we do not require convexity of the Hamiltonian, thus extending the result
of \cite{mfgmain} to a considerably broader class of Hamiltonians.
