In 1883, S. Lie found 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.

Ibragimov, N., Magri, F. (2004). Geometric proof of Lie's linearization theorem. NONLINEAR DYNAMICS, 36(1), 41-46 [10.1023/B:NODY.0000034645.77245.26].

Geometric proof of Lie's linearization theorem

MAGRI, FRANCO
2004

Abstract

In 1883, S. Lie found 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.
Articolo in rivista - Articolo scientifico
Nonlinear Dynamics,Lie Theory
English
2004
36
1
41
46
none
Ibragimov, N., Magri, F. (2004). Geometric proof of Lie's linearization theorem. NONLINEAR DYNAMICS, 36(1), 41-46 [10.1023/B:NODY.0000034645.77245.26].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/11454
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