On the Cessation of Motion in a Problem with Coulomb Friction

We consider the problem of motion of a heavy particle on a rough horizontal plane. The particle is subject to the Coulomb friction force and an attractive force proportional to the distance to the attracting center. Such a force can be caused by a linear elastic spring. This note is devoted to answering the following question: Is it true that the particle stops in finite time for any initial conditions? Despite the fact that an affirmative answer is physically obvious, the formal proof of this fact turns out to be surprisingly complicated. Ultimately, everything reduces to the study of the asymptotic behavior of solutions to a smooth nonlinear system in the neighborhood of a degenerate equilibrium position. In addition, we formulate an open problem.