By introducing unknown Sign-like fields of Level-Sets type, the Signorini-Moreau dynamic contact conditions are set merely as boundary equations. From this setting, a continuous hybrid weak–strong formulation for dynamic contact between deformable solids is derived and a new Lagrangian formulation (we call stabilized) generalizing both the classical and augmented ones is obtained. Friction phenomena are treated similarly. In the global problem, the irregular Sign-like fields stand for the intrinsic contact unknown ones. This problem is discretized by means of time, space and collocation schemes. Some numerical experimentations are carried out, showing the potential of our developments.