Variational principles for differential equations with symmetries and conservation laws. I: Second order scalar equations
type: Item
instance of: scholarly article
title: Variational principles for differential equations with symmetries and conservation laws. I: Second order scalar equations
zbMATH Open document ID: 0804.49030
DOI: 10.1007/BF01459780
author: I. M. Anderson
author: Juha Pohjanpelto
published in: Mathematische Annalen
publication date: 1994-11-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165205
review text: The Noether correspondence between the symmetries of a variational problem and conservation laws for the Euler-Lagrange equations suggests the following general problem. Suppose a system of differential equations admits a large group of symmetries and that each one of these symmetries generates a conservation law for the equations. Must these differential equations be the Euler-Lagrange equations for some problem in the calculus of variations? In 1977, F. Takens formulated this problem rigorously and answered it affirmatively in several special cases. In this paper and its sequel we generalize a number of Takens' results. Here we undertake a detailed study of this problem for the case of second order scalar differential equations. Let \(\Gamma\) be a given infinitesimal Lie transformation group acting on the space of independent and dependent variables. We show that if a second order scalar differential equation \(\Delta=0\) is \(\Gamma\) invariant and admits \(\Gamma\) conservation laws, then the Helmholtz operator associated to \(\Delta\) must satisfy certain differential conditions. A careful analysis of these differential equations allows us to substantially generalize Takens' result for second order scalar differential equations. For a particular class of infinitesimal Lie groups \(\Gamma\), we are able to completely reduce the solution of the differential conditions on the Helmholtz operator to the solution of a simple system of equations depending only on the structure constants of \(\Gamma\). We illustrate the interplay between symmetries, conservation laws and variational principles for second order scalar partial differential equations with many examples.
reviewed by: Hitoshi Abe
Mathematics Subject Classification ID: 49N45
Mathematics Subject Classification ID: 58D19
Mathematics Subject Classification ID: 58E30
zbMATH DE Number: 683140
zbMATH Keywords: Gelfand-Dickey transform
zbMATH Keywords: conservation laws
zbMATH Keywords: inverse problem
zbMATH Keywords: symmetries of a variational problem
zbMATH Keywords: Euler-Lagrange equations
zbMATH Keywords: second order scalar differential equations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 683140