The Geometry of the Painlevé Paradox

Painlevé showed that there can be inconsistency and indeterminacy in solutions to the equations of motion of a two-dimensional (2D) rigid body moving on a sufficiently rough surface. The study of Painlevé paradoxes in three dimensions (3D) has received far less attention. In this paper, we highlight the pivotal role in the dynamics of the azimuthal angular velocity ψ by proving the existence of three critical values of ψ, 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.