  We consider deterministic mean field games where the dynamics of a typical
agent is non-linear with respect to the state variable and affine with respect
to the control variable. Particular instances of the problem considered here
are mean field games with control on the acceleration. We focus our attention
on the approximation of such mean field games by analogous problems in discrete
time and finite state space. For these approximations, we show the existence
and, under an additional monotonicity assumption, uniqueness of solutions. In
our main result, we establish the convergence of equilibria of the discrete
mean field games problems towards equilibria of the continuous one. Finally, we
provide some numerical results for two MFG problems. In the first one, the
dynamics of a typical player is nonlinear with respect to the state and, in the
second one, a typical player controls its acceleration.
