Geometric proof of Lie's linearization theorem

Document type: Journal Articles
Article type: Original article
Peer reviewed: Yes
Author(s): Nail H. Ibragimov, F. Magri
Title: Geometric proof of Lie's linearization theorem
Translated title: Geometric proof of Lie's linearization theorem
Journal: Nonlinear Dynamics
Year: 2004
Volume: 36
Issue: 1
Pagination: 41-46
ISSN: 0924-090X
Publisher: Springer
City: Dordrecht
ISI number: 000222611300005
Organization: Blekinge Institute of Technology
Department: School of Engineering - Dept. Mathematics and Science (Sektionen för teknik – avd. för matematik och naturvetenskap)
School of Engineering S- 371 79 Karlskrona
+46 455 38 50 00
Authors e-mail:
Language: English
Abstract: S. Lie found in 1883 the general form of all second-order ordinary differential equations transformable to the linear equation by a change of variables and proved that their solution reduces to integration of a linear third-order ordinary differential equation. He showed that the linearizable equations are at most cubic in the first-order derivative and described a general procedure for constructing linearizing transformations by using an over-determined system of four equations. We present here a simple geometric proof of the theorem, known as Lie's linearization test, stating that the compatibility of Lie's four auxiliary equations furnishes a necessary and sufficient condition for linearization.
Subject: Mathematics\Analysis
Keywords: Lie's linearization test, symmetries, linearizable equations, Lie group analysis, differential equations