Towards an algebro-geometric interpretation of the Neumann system
type: Item
instance of: scholarly article
title: Towards an algebro-geometric interpretation of the Neumann system
zbMATH Open document ID: 0582.35102
author: H. Flaschka
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1984-01-01T00:00:00Z
review text: The author considers a point moving on the sphere \(\xi^ 2_ 1+...+\xi^ 2_ n=1\) under the influence of a quadratic potential \(\phi (\xi)=-\sum x_ i\xi^ 2_ i.\) The differential equation describing this motion is \({\ddot \xi}{}_ i+q\xi_ i=x_ i\xi_ i\), \(q=\sum \xi^ 2_ i+x_ i\xi^ 2_ i\), \(\xi^ 2_ 1+...+\xi^ 2_ n=1\) (Neumann's system). This problem was solved via (ultra-elliptic) quadratures by Neumann for \(n=3\). Neumann's system is closely related to the Burchnall-Chaundy-Krichever theory of second order differential operators \(D^ 2+q(t)\). The author derives from this theory Lax equations and constants of motion for Neumann's system.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 14H99
zbMATH DE Number: 3931606
zbMATH Keywords: affine curve linebundle
zbMATH Keywords: quadratic potential
zbMATH Keywords: Neumann's system
zbMATH Keywords: Burchnall-Chaundy-Krichever theory
zbMATH Keywords: Lax equations
zbMATH Keywords: constants of motion
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228807
OpenAlex ID: W1996834583
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DOI: 10.2748/TMJ/1178228807
description: scientific article; zbMATH DE number 3931606