Abstract: I will talk about a joint research project with Y. Achdou, C. Marchi and N. Tchou about some models of deterministic MFGs where the Hamiltonian is not coercive in the gradient term because the dynamics of the generic player must fulfill some constraints, holonomic or non-holonomic, or fail to be controllable.First of all I will outline the model where the generic player can move in the whole space but it has some forbidden directions. Afterwards, I will treat the case where the dynamic of the generic agent is controlled by the acceleration; I consider both the case where the agents can move in the whole space and the case where the agents are constrained to remains in a given bounded region. For the first models we study the existence of weak solutions and their representation formula, while for the constrained MFG model controlled by the acceleration we study the existence of relaxed equilibria in the Lagrangian setting which are described by a probability measure on optimal trajectories.