Painlevé paradox

Counterexample

Consider a rigid body that touches a fixed rigid surface at one point and slides on it. The contact is unilateral: the normal contact force $\lambda$ satisfies $\lambda \ge 0$, and the body may leave the surface but not enter it. The tangential contact force obeys Coulomb's law of friction with coefficient $\mu > 0$: during sliding it has size $\mu \lambda$ and opposes the slip velocity. Let $y$ be the height of the contact point above the surface. While the contact point slides, Newton's laws give $$\ddot y = A + B \lambda.$$ Here $A$ depends on the positions and velocities. The coefficient $B$ depends on the positions, on the direction of slip and on $\mu$. The contact conditions are complementarity conditions: $$\lambda \ge 0, \qquad \ddot y \ge 0, \qquad \lambda\, \ddot y = 0.$$ If $B > 0$, there is exactly one solution: $\lambda = 0$ when $A \ge 0$ and $\lambda = -A/B$ when $A < 0$. If $B < 0$ and $A < 0$, there is no solution: $\lambda = 0$ gives $\ddot y = A < 0$, and $\lambda > 0$ gives $\ddot y < 0$. If $B < 0$ and $A > 0$, there are two solutions: $\lambda = 0$, and $\lambda = -A/B > 0$ with $\ddot y = 0$. For large enough $\mu$ the coefficient $B$ becomes negative in some positions. Then the rigid-body equations with Coulomb friction can have no solution or several solutions.