  In the present work, we study deterministic mean field games (MFGs) with
finite time horizon in which the dynamics of a generic agent is controlled by
the acceleration. They are described by a system of PDEs coupling a continuity
equation for the density of the distribution of states (forward in time) and a
Hamilton-Jacobi (HJ) equation for the optimal value of a representative agent
(backward in time). The state variable is the pair $(x, v)\in R^N\times R^N$
where x stands for the position and v stands for the velocity. The dynamics is
often referred to as the double integrator. In this case, the Hamiltonian of
the system is neither strictly convex nor coercive, hence the available results
on MFGs cannot be applied. Moreover, we will assume that the Hamiltonian is
unbounded w.r.t. the velocity variable v. We prove the existence of a weak
solution of the MFG system via a vanishing viscosity method and we characterize
the distribution of states as the image of the initial distribution by the flow
associated with the optimal control.
