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Title: Generalised Eisenhart lift of the Toda chain

Abstract

The Toda chain of nearest neighbour interacting particles on a line can be described both in terms of geodesic motion on a manifold with one extra dimension, the Eisenhart lift, or in terms of geodesic motion in a symmetric space with several extra dimensions. We examine the relationship between these two realisations and discover that the symmetric space is a generalised, multi-particle Eisenhart lift of the original problem that reduces to the standard Eisenhart lift. Such generalised Eisenhart lift acts as an inverse Kaluza-Klein reduction, promoting coupling constants to momenta in higher dimension. In particular, isometries of the generalised lift metric correspond to energy preserving transformations that mix coordinates and coupling constants. A by-product of the analysis is that the lift of the Toda Lax pair can be used to construct higher rank Killing tensors for both the standard and generalised lift metrics.

Authors:
 [1];  [2]
  1. DEFIS, Universidade Federal de Ouro Preto, Campus Morro do Cruzeiro, 35400-000 Ouro Preto, MG (Brazil)
  2. DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)
Publication Date:
OSTI Identifier:
22251209
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 2; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COORDINATES; COUPLING CONSTANTS; KALUZA-KLEIN THEORY; MATHEMATICAL SPACE; METRICS; STANDARDS; SYMMETRY; TENSORS; TRANSFORMATIONS

Citation Formats

Cariglia, Marco, E-mail: marco@iceb.ufop.br, and Gibbons, Gary, E-mail: g.w.gibbons@damtp.cam.ac.uk. Generalised Eisenhart lift of the Toda chain. United States: N. p., 2014. Web. doi:10.1063/1.4866318.
Cariglia, Marco, E-mail: marco@iceb.ufop.br, & Gibbons, Gary, E-mail: g.w.gibbons@damtp.cam.ac.uk. Generalised Eisenhart lift of the Toda chain. United States. doi:10.1063/1.4866318.
Cariglia, Marco, E-mail: marco@iceb.ufop.br, and Gibbons, Gary, E-mail: g.w.gibbons@damtp.cam.ac.uk. Sat . "Generalised Eisenhart lift of the Toda chain". United States. doi:10.1063/1.4866318.
@article{osti_22251209,
title = {Generalised Eisenhart lift of the Toda chain},
author = {Cariglia, Marco, E-mail: marco@iceb.ufop.br and Gibbons, Gary, E-mail: g.w.gibbons@damtp.cam.ac.uk},
abstractNote = {The Toda chain of nearest neighbour interacting particles on a line can be described both in terms of geodesic motion on a manifold with one extra dimension, the Eisenhart lift, or in terms of geodesic motion in a symmetric space with several extra dimensions. We examine the relationship between these two realisations and discover that the symmetric space is a generalised, multi-particle Eisenhart lift of the original problem that reduces to the standard Eisenhart lift. Such generalised Eisenhart lift acts as an inverse Kaluza-Klein reduction, promoting coupling constants to momenta in higher dimension. In particular, isometries of the generalised lift metric correspond to energy preserving transformations that mix coordinates and coupling constants. A by-product of the analysis is that the lift of the Toda Lax pair can be used to construct higher rank Killing tensors for both the standard and generalised lift metrics.},
doi = {10.1063/1.4866318},
journal = {Journal of Mathematical Physics},
number = 2,
volume = 55,
place = {United States},
year = {Sat Feb 15 00:00:00 EST 2014},
month = {Sat Feb 15 00:00:00 EST 2014}
}
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