Two remarks about the connection of Jacobi and Neumann integrable systems
type: Item
instance of: scholarly article
title: Two remarks about the connection of Jacobi and Neumann integrable systems
zbMATH Open document ID: 0815.58011
DOI: 10.1007/BF02572325
published in: Mathematische Zeitschrift
publication date: 1994-12-21T00:00:00Z
full work available at URL: https://eudml.org/doc/174654
review text: The paper can be considered as the addendum to the papers by \textit{H. Knörrer} [J. Reine Angew. Math. 334, 69-78 (1982; Zbl 0478.58014)] and \textit{J. Moser} [Proc. Math. 8, 233-290 (1980; Zbl 0468.58011)] and consists of two remarks about the connection of Jacobi and Neumann systems. H. Knörrer [loc. cit.] discovered that the geodesics on the ellipsoid can be transformed into the trajectories of the motion of a point on the unit sphere under the influence of certain quadratic potential (Neumann system). The first remark says that Knörrer's theorem can be extended to the case of the motion on the ellipsoid in the potential field with \(U(x) = {1\over 2} \varepsilon x^ 2\). The second remark is an interpretation of the Lagrange multiplier for the geodesic flow on the ellipsoid in terms of the stability of the equilibrium point for a one-dimensional mechanical problem.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 58E99
zbMATH DE Number: 704921
zbMATH Keywords: Jacobi problem
zbMATH Keywords: potential field
zbMATH Keywords: Neumann system
zbMATH Keywords: geodesics
zbMATH Keywords: Lagrange multiplier
author: A. P. Veselov
MaRDI profile type: Publication
cites work: NON-LINEAR EQUATIONS OF KORTEWEG-DE VRIES TYPE, FINITE-ZONE LINEAR OPERATORS, AND ABELIAN VARIETIES
cites work: Geodesics on quadrics and a mechanical problem of C. Neumann.
description: scientific article; zbMATH DE number 704921