On particles in equilibrium on the real line
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- by Agelos Georgakopoulos and Mihail N. Kolountzakis PDF
- Proc. Amer. Math. Soc. 145 (2017), 3501-3511 Request permission
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
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Additional Information
- Agelos Georgakopoulos
- Affiliation: Mathematics Institute, University of Warwick, CV4 7AL, Coventry, United Kingdom
- MR Author ID: 805415
- ORCID: 0000-0001-6430-567X
- 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
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3501-3511
- MSC (2010): Primary 70C20
- DOI: https://doi.org/10.1090/proc/13492
- MathSciNet review: 3652802