The Geometry of the Painlevé paradox

Noah Cheesman; S. John Hogan; Kristian Uldall Kristiansen

The Geometry of the Painlevé paradox

Noah Cheesman, S. John Hogan, Kristian Uldall Kristiansen

Research output: Working paper › Preprint

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Abstract

Painlevé 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 Painlevé 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.

Original language	English
Number of pages	27
Publication status	Unpublished - 27 Oct 2021

Research Groups and Themes

Engineering Mathematics Research Group

Keywords

math.DS
37N15, 70E18, 74H20, 74H25
Painlevé paradox
geometry
mechanics

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This is a pre-print version of the article. It first appeared online via arXiv at https://arxiv.org/abs/2110.14324. Please refer to any applicable terms of use of the publisher.
Submitted manuscript, 1.74 MBLicence: Unspecified

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Applications of geometric singular perturbation theory to mechanics: the Painlevé paradox in 3D and related problems
Cheesman, N. (Author), Hogan, J. (Supervisor) & Kristiansen, K. U. (Supervisor), 21 Jun 2022
Student thesis: Doctoral Thesis › Doctor of Philosophy (PhD)

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KW - geometry

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BT - The Geometry of the Painlevé paradox

ER -

The Geometry of the Painlevé paradox

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Research Groups and Themes

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Applications of geometric singular perturbation theory to mechanics: the Painlevé paradox in 3D and related problems

Cite this