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

Short time solution to the master equation of a first order mean field game system