On singular points of equations of mechanics

A system of ordinary differential equations whose right-hand side has no limit at some singular point is considered. This situation is typical of mechanical systems with Coulomb friction in a neighborhood of equilibrium. The existence and uniqueness of solutions to the Cauchy problem is analyzed. A key property is that the phase curve reaches the singular point in a finite time. It is shown that the subsequent dynamics depends on the extension of the vector field to the singular point according to the physical interpretation of the problem: systems coinciding at all point, except for the singular one, can have different solutions. Uniqueness conditions are discussed.