An alternative approach to the Painlevé paradox through constitutive characterization of constraints in impulsive Mechanics

TL;DR

The paper reframes the classical Painlevé paradox within Regular Geometric Impulsive Mechanical Systems (RGIMS), modeling friction as an internal kinetic constraint and introducing an impulsive constitutive rule to restore determinism. The core idea is a two-parameter impulse $I(P_L)=σ K_S + β K_B$ that determines the outgoing velocity from the incoming state at impact, with $σ$ and $β$ depending on mass, geometry, and potentially material properties. Through three illustrative examples, it shows how appropriate choices of $σ$ and $β$ yield rebound, braking, or detachment, aligning theoretical predictions with experimental observations and enabling direct coefficient determination. The approach offers a general, experimentally testable framework for impulsive constrained systems and points to broad extensions, including 3D Painlevé problems and inert constraints relevant to robotics and biomechanics.

Abstract

We frame the Painlevè mechanical system, which has been extensively studied because of the paradox it generates, within the class of Regular Geometric Impulsive Mechanical Systems (RGIMS), by modeling it as a mechanical system subject to a rough unilateral positional constraint $\cal{S}$, where friction is represented by an instantaneous kinetic constraint $\cal{B}$, internal to $\cal{S}$ and of impulsive nature. The evolution of the system is therefore determined by the choice of a constitutive characterization for these constraints, a choice that restores mechanical determinism and eliminates any paradoxical aspects of the system's behavior, in agreement with experimental evidence. It is shown that, similarly to what occurs in general non ideal impulsive systems, the choice of a constitutive characterization of the constraint system depends on the determination of two numerical coefficients $σ$ and $β$, which depend on the kinematic and mass-related data of the system, and possibly also on physical quantities not strictly of a mechanical nature, such as material properties. The simplicity of the model also allows for a straightforward experimental analysis of the system's behavior and for the experimental determination of the values of these coefficients.