The Geometry of the Painlev\'e paradox

TL;DR

This paper explores the geometric structure of the Painlevé paradox in 3D rigid body dynamics, revealing critical angular velocities and explaining complex behaviors through geometric analysis.

Contribution

It extends the understanding of Painlevé paradoxes from 2D to 3D, identifying critical angular velocities and uncovering the geometric richness of the 3D problem.

Findings

Existence of three critical azimuthal angular velocities in 3D

The 2D problem is highly singular

Geometric analysis explains recent numerical results

Abstract

Painlev\'e showed that there can be inconsistency and indeterminacy in solutions to the equations of motion of a 2D rigid body moving on a sufficiently rough surface. The study of Painlev\'e paradoxes in 3D has received far less attention. In this paper, we highlight the pivotal role in the dynamics of the azimuthal angular velocity Psi by proving the existence of three critical values of Psi, one of which occurs independently of any paradox. We show that the 2D problem is highly singular and uncover a rich geometry in the 3D problem which we use to explain recent numerical results.