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 http://www.tek.bth.se/ |
| Authors e-mail: | nib@bth.se |
| 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 |
