Group invariant solutions without transversality
type: Item
instance of: scholarly article
title: Group invariant solutions without transversality
zbMATH Open document ID: 0961.35007
author: I. M. Anderson
author: Mark E. Fels
author: C. G. Torre
published in: Communications in Mathematical Physics
publication date: 2001-05-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/math-ph/9910014
full work available at URL: https://arxiv.org/abs/math-ph/9910015
full work available at URL: http://citeseerx.ist.psu.edu/viewdoc/summary
review text: In this paper the authors find the group invariant solutions of a system of partial differential equations by the generalized Lie method. It is described a general method for characterizing the group invariant sections of a definite bundle. It is well known that, if the local transversality condition fails, then the general solution to the infinitesimal invariance equation will depend upon fewer arbitrary functions than the original number of dependent variables. It leads us to the reduced systems of differential equations with both fewer independent and dependent variables. The kinematic and dynamic reduction diagrams for the action of a finite-dimensional Lie group on the space of independent and dependent variables are defined. The main properties of these diagrams are pointed out. The authors find the kinematic and dynamic reduction diagrams for the group invariant solutions for some specific systems of differential equations applicable in mathematical physics and differential geometry. There are considered the reductions of the harmonic map equations the Euler equations for incompressible fluid flow. Also on some examples from mechanics the authors show that there is no guarantee that the reduced equations for the group invariant solutions posses any natural inherited Hamiltonian formulation.
reviewed by: Violeta Tretynyk
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 58J70
zbMATH DE Number: 1523450
zbMATH Keywords: Lie group
zbMATH Keywords: kinematic and dynamic reduction diagrams
zbMATH Keywords: harmonic map equations
zbMATH Keywords: Euler equations for incompressible fluid flow
zbMATH Keywords: reduction
MaRDI profile type: Publication
OpenAlex ID: W2951935963
arXiv ID: math-ph/9910015
DOI: 10.1007/S002200000215
description: scientific article