Ordinary differential equations invariant under translation in the independent variable and rescaling: The Lagrangian formulation
type: Item
instance of: scholarly article
title: Ordinary differential equations invariant under translation in the independent variable and rescaling: The Lagrangian formulation
zbMATH Open document ID: 1088.34033
author: S. Moyo
author: P. G. L. Leach
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-06-01T00:00:00Z
review text: Second-order ordinary differential equations invariant under translation \(\partial_x\) in the independent variable and rescaling \(-qx \partial_x +y \partial_y\) are studied from the Lagrangian point of view. The general form of such equation is \[ \frac{y''}{y^{2q+1}} +f(\frac{y'}{y^{q+1}})=0. \tag{1} \] The authors present a method to determine the Lagrangian of equation (1) and to analyze the equations in terms of the Painlevé analysis and their Lagrangians from the Noetherian approach. The establishment of a relationship between the Euler-Lagrange equations and their Lagrangians provides a general method for dealing with equations of the class under consideration. The Lagrangian of equation (1) is represented in the form \(L=y^n F(y'/y^2)\) since the variable \(y'/y^2\) appears as an arbitrary argument in the second-order differential equation. The multiplier of \(F\) in the expression of the Lagrangian can be chosen to be any function of \(y\) or \(y'\). However, the differential equation for \(F\) imposes an effective restriction on the form of the multiplier of \(F\) due to the problems of integrating the equation. The associated Euler-Lagrange equation is \((2u^2+f(u))F'' -nuF'+nF=0\) with \(u=y'/y^2\). The coefficient of \(F''\) in this equation is taken to be \(2u^2+f(u)=m(u+\alpha)^2+\beta\). A feature associated with the class of differential equations for \(F\) resulting from the choice of a quadratic \(f=f(u)\) is that in general they are hypergeometric functions. The Painlevé analysis of the equations arising in the different cases is performed.
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 70H99
zbMATH DE Number: 2173084
zbMATH Keywords: second-order ordinary differential equations
zbMATH Keywords: rescaling
zbMATH Keywords: Lagrangian
zbMATH Keywords: Painlevé analysis
zbMATH Keywords: self-similarity
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.10.005
OpenAlex ID: W2017039690
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DOI: 10.1016/J.JMAA.2004.10.005
description: scientific article; zbMATH DE number 2173084