N-body dynamics on closed surfaces: the axioms of mechanics

Stefanella Boatto; David Gerard Dritschel; Rodrigo G Schaefer

doi:10.1098/rspa.2016.0020

N-body dynamics on closed surfaces: the axioms of mechanics

Stefanella Boatto, David Gerard Dritschel, Rodrigo G Schaefer

Research output: Contribution to journal › Article › peer-review

Abstract

A major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of N-body and N-vortex dynamics on Riemann surfaces. In this paper, we show how the two problems are, in fact, closely related when considering the role played by the intrinsic geometry of the surface. This enables a straightforward deduction of the dynamics of point masses, using recently derived results for point vortices on general closed differentiable surfaces M endowed with a metric g. We find, generally, that Kepler's Laws do not hold. What is more, even Newton's First Law (the law of inertia) fails on closed surfaces with variable curvature (e.g. the ellipsoid).

Original language	English
Article number	20160020
Pages (from-to)	1-20
Number of pages	20
Journal	Proceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences
Volume	472
Issue number	2192
Early online date	1 Aug 2016
DOIs	https://doi.org/10.1098/rspa.2016.0020
Publication status	Published - Aug 2016

Keywords

N-body problem
Point-vortex dynamics
Hamiltonian systems
Newton's Laws
Surfaces of revolution
Equivalence principle

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10.1098/rspa.2016.0020

Boatto_2016_N-body_PRSA_AAM
© 2016, the Author(s). Published by the Royal Society. All rights reserved. This work is made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://dx.doi.org/10.1098/rspa.2016.0020
Accepted author manuscript, 1.9 MB

Geophysical Vortices: The Structure, stability and interaction of geophysical vortices
Reinaud, J. N. (PI), Dritschel, D. G. (CoI) & Scott, R. K. (CoI)
EPSRC
5/01/10 → 1/11/13
Project: Standard

David Gerard Dritschel
- david.dritschelst-andrews.acuk
- Applied Mathematics - Professor
- Marine Alliance for Science & Technology Scotland
Person: Academic

Cite this

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title = "N-body dynamics on closed surfaces: the axioms of mechanics",

abstract = "A major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of N-body and N-vortex dynamics on Riemann surfaces. In this paper, we show how the two problems are, in fact, closely related when considering the role played by the intrinsic geometry of the surface. This enables a straightforward deduction of the dynamics of point masses, using recently derived results for point vortices on general closed differentiable surfaces M endowed with a metric g. We find, generally, that Kepler's Laws do not hold. What is more, even Newton's First Law (the law of inertia) fails on closed surfaces with variable curvature (e.g. the ellipsoid).",

keywords = "N-body problem, Point-vortex dynamics, Hamiltonian systems, Newton's Laws, Surfaces of revolution, Equivalence principle",

author = "Stefanella Boatto and Dritschel, {David Gerard} and Schaefer, {Rodrigo G}",

note = "D.G.D. gratefully acknowledges support for this research from CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico ) and FINEP (Inova{\c c}{\~a}o e Pesquisa) in Brazil, and from the UK Engineering and Physical Sciences Research Council (grant no. EP/H001794/1)",

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doi = "10.1098/rspa.2016.0020",

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KW - Newton's Laws

KW - Surfaces of revolution

KW - Equivalence principle

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N-body dynamics on closed surfaces: the axioms of mechanics

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Geophysical Vortices: The Structure, stability and interaction of geophysical vortices

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David Gerard Dritschel

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