Geometrical properties of the multidimensional nonlinear differential equations and the Finsler metrics of phase spaces of dynamical systems
type: Item
instance of: scholarly article
title: Geometrical properties of the multidimensional nonlinear differential equations and the Finsler metrics of phase spaces of dynamical systems
zbMATH Open document ID: 0856.53024
DOI: 10.1007/BF01016138
published in: Theoretical and Mathematical Physics
publication date: 1997-02-24T00:00:00Z
review text: The first part of the paper is devoted to describing some nonlinear differential equations arising as conditions on the curvature tensors of 3- and 4-dimensional Riemannian manifolds. The metrics \(ds\) are assumed to be of the standard forms \(ds^2 = \Phi^2dt^2 + A^2dx^2 + B^2dy^2 + C^2dz^2\), etc. In the second part some questions of the Cartan theory for equations \(y''=f(x,y,y')\) and geometrical properties of the space of linear elements \((x,y,y')\) are discussed and examples of Finsler metrics for the second order differential equation \(y''=3(y')^2/y + (1/x-my)y'+ (nx^3-1/x)y^4 + (nx^2+t) y^3-sy^2/x\) are constructed. This equation is equivalent to E. Lorentz's nonlinear dynamical system \(\dot x=k(x-y)\), \(\dot y=rx-y-zx\), \(\dot z=xy-bz\). Those examples are found from special Finsler metrics \(ds= [a(x,y)dy + b(x,y) dx]^r dx^s\), \(r+s=1\), which give rise to Berwald spaces with the main scalar \(I\) satisfying \(I^2>4\).
Mathematics Subject Classification ID: 53B40
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 58A15
Mathematics Subject Classification ID: 70G10
zbMATH DE Number: 907304
zbMATH Keywords: nonlinear differential equations
zbMATH Keywords: curvature tensors
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: Cartan theory
zbMATH Keywords: Finsler metrics
zbMATH Keywords: Berwald spaces
author: Valery S. Dryuma
reviewed by: Makoto Matsumoto
MaRDI profile type: Publication
cites work: Construction of higher-dimensional nonlinear integrable systems and of their solutions
cites work: Q4304304
description: scientific article; zbMATH DE number 907304