I will talk about a research project with Y. Achdou, A. Cutri', C. Marchi and N. Tchou about some models of degenerate deterministic Mean Field Games (MFG). These games are modelized by a system of two partial differential equations: a backward in time Hamilton-Jacobi equation and a forward in time continuity equation. In these cases the Hamiltonian is not strictly convex in the gradient term because the dynamics of the generic player is controlled by the acceleration or it has some forbidden directions. I will consider both the cases without and with state constraints. Our results concern some properties of the associated optimal control problems which will play a crucial role for obtaining the existence of weak solutions for the MFG system. For the constrained case we study the representation of the solution by means of relaxed equilibria in the Lagrangian setting which are described by a probability measure on optimal trajectories.