An application of geodesic modeling of second-order differential equations
type: Item
instance of: scholarly article
title: An application of geodesic modeling of second-order differential equations
zbMATH Open document ID: 0624.58008
DOI: 10.1007/BF01163712
author: V. A. Igoshin
author: Ya. L. Shapiro
author: E. I. Yakovlev
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: It is proved that for the second-order ordinary differential equation (generalizing the electromagnetic field equation \[ (\nabla /dt)(m(x)dx/dt)=(2e/c)F^{\#}(x,dx/dt)-(e^ 2/2c^ 2m(x))\text{grad}(1/f) \] for any points \(p,q\in M\) not conjugate, in a Riemannian manifold M, in every homotopy class there exists a continuum of distinct solutions joining p with q. Here \(F^{\#}\) is defined by \(F(X,Y)=g(X,F^{\#}Y)\) for g a complete Riemannian metric (\(\nabla\) is its affine Levi-Civita connection), F a closed 2-form on M, m,f function on M, \(f>0\). The idea is to lift the equation to a geodesic one in a higher dimensional manifold (a geodesic modeling). The reader will find a lot of references to applications in physics.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 78A35
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 4013571
zbMATH Keywords: second-order ordinary differential equation
zbMATH Keywords: electromagnetic field equation
zbMATH Keywords: geodesic modeling
MaRDI profile type: Publication
cites work: Q3042051
cites work: Q5760260
cites work: Q5731354
cites work: Q5638376
full work available at URL: https://doi.org/10.1007/bf01163712
OpenAlex ID: W2073165583
description: scientific article; zbMATH DE number 4013571