# On The Well-Posedness of Painlevé’s Example

## Abstract

This note aims at illustrating with a simple example the problems of wellposedness of the dynamics of mechanical systems with unilateral constraints and dry friction. The analysed example is the wellknown Painlevé system (Painlevé, 1895). In particular we will focus on singularities of the dynamics in sliding regimes, i.e. configurations at which the contact force diverges to infinity. The problem of inconsistencies, that is configurations for which no continuous solution exists, will also be examined, as well as indeterminacies, i.e. configurations which lead to non-uniqueness of solutions. More precisely, there may be no bounded contact forces that permit the satisfaction of the unilateral constraints. Consequently the space within which solutions have to be defined and found must be augmented by discontinuous velocities and distributional interaction forces. For instance, some sort of Impact Without Collisions (IW/OC) can be introduced when dry friction is present. This is a phenomenon such that velocity jumps can occur with zero initial normal velocity, primarily due to Amontons-Coulomb friction. It is related to Kilmister’s principle of constraints: “a unilateral constraint must be verified with (bounded) forces each time it is possible, and with impulses if and only if it is not possible with bounded forces”. Therefore this a priori stated principle tells us that if one is able to exhibit dynamical situations for which a bounded force cannot be found such that the contraints are satisfied, then one may use an impulsive force at the contact point. Concerning non-uniqueness of solutions, Painlevé also proposes his principle: “two rigid bodies, which under given conditions would not produce any pressure on one another, if they were ideally smooth, would likewise not act on one another if they were rough”. These two principles have not been given any experimental validation to the best of our knowledge.

## Keywords

Singular Point Contact Force Linear Complementarity Problem Critical Line Unilateral Constraint## Preview

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