The Geometry of the Painlevé paradox

Noah Cheesman; S J Hogan; K.U. Kristiansen

doi:10.48550/arXiv.2110.14324

The Geometry of the Painlevé paradox

Noah Cheesman, S J Hogan, K.U. Kristiansen

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

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.

Original language	English
Pages (from-to)	1798-1831
Number of pages	27
Journal	SIAM Journal on Applied Dynamical Systems
Volume	21
Issue number	3
Early online date	13 Jul 2022
DOIs	https://doi.org/10.48550/arXiv.2110.14324 https://doi.org/10.1137/21M1455590
Publication status	Published - 1 Sept 2022

Bibliographical note

Publisher Copyright:
© 2022, Society for Industrial and Applied Mathematics.

Research Groups and Themes

Engineering Mathematics Research Group

Keywords

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

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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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title = "The Geometry of the Painlev{\'e} paradox",

abstract = "Painlev{\'e} 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 Painlev{\'e} 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.",

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doi = "10.48550/arXiv.2110.14324",

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T1 - The Geometry of the Painlevé paradox

AU - Cheesman, Noah

AU - Hogan, S J

AU - Kristiansen, K.U.

PY - 2022/9/1

Y1 - 2022/9/1

N2 - 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.

AB - 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.

KW - math.DS

KW - 37N15, 70E18, 74H20, 74H25

KW - Painlevé paradox

KW - geometry

KW - mechanics

U2 - 10.48550/arXiv.2110.14324

DO - 10.48550/arXiv.2110.14324

M3 - Article (Academic Journal)

SN - 1536-0040

VL - 21

SP - 1798

EP - 1831

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

IS - 3

ER -

The Geometry of the Painlevé paradox

Abstract

Bibliographical note

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Keywords

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Student theses

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

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