Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

On particles in equilibrium on the real line


Authors: Agelos Georgakopoulos and Mihail N. Kolountzakis
Journal: Proc. Amer. Math. Soc. 145 (2017), 3501-3511
MSC (2010): Primary 70C20
DOI: https://doi.org/10.1090/proc/13492
Published electronically: February 15, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study equilibrium configurations of infinitely many identical particles on the real line or finitely many particles on the circle, such that the (repelling) force they exert on each other depends only on their distance. The main question is whether each equilibrium configuration needs to be an arithmetic progression. Under very broad assumptions on the force we show this for the particles on the circle. In the case of infinitely many particles on the line we show the same result under the assumption that the maximal (or the minimal) gap between successive points is finite (positive) and assumed at some pair of successive points. Under the assumption of analyticity for the force field (e.g. the Coulomb force) we deduce some extra rigidity for the configuration: knowing an equilibrium configuration of points in a half-line determines it throughout. Various properties of the equlibrium configuration are proved.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 70C20

Retrieve articles in all journals with MSC (2010): 70C20


Additional Information

Agelos Georgakopoulos
Affiliation: Mathematics Institute, University of Warwick, CV4 7AL, Coventry, United Kingdom

Mihail N. Kolountzakis
Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, Voutes Campus, GR-700 13, Heraklion, Crete, Greece
Email: kolount@gmail.com

DOI: https://doi.org/10.1090/proc/13492
Received by editor(s): June 14, 2016
Received by editor(s) in revised form: September 18, 2016
Published electronically: February 15, 2017
Additional Notes: The first author was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 639046).
The second author has been partially supported by the “Aristeia II” action (Project FOURIERDIG) of the operational program Education and Lifelong Learning and is co-funded by the European Social Fund and Greek national resources.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2017 American Mathematical Society