New results on Painleve paradoxes

In this note, we deal with the dynamics of a mechanical system subject to unilateral constraints. In particular, we shall focus on the integration of such a system that is known to be closely related to so-called linear complementary problem (LCP). Except a in very simple case like codimension 1, frictionless constraints, the problem of well-posedness (existence and uniqueness of solution) to such hybrid dynamic systems (smooth dynamics + LCP + shock dynamics) is a big challenge. We concentrate on the well-known Painleve example the dynamics of which in a sliding regime may be singular, depending on the friction coefficient. A new critical friction coefficient is presented below, the contact forces of which remain bounded. Moreover, a detailed analysis of the vector field near the singularity shows that the eventual divergence of the contact forces does not call into question the well-posedness of the model. (C) Elsevier, Paris.