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Symmetry 2016, 8(3), 15; https://doi.org/10.3390/sym8030015

Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

Instituto de Matemática Interdisciplinar and Depto. Geometría y Topología, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid E-28040, Spain
Academic Editor: Roman M. Cherniha
Received: 1 December 2015 / Revised: 24 February 2016 / Accepted: 26 February 2016 / Published: 17 March 2016
(This article belongs to the Special Issue Symmetry and Integrability)
Full-Text   |   PDF [254 KB, uploaded 17 March 2016]

Abstract

A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems. View Full-Text
Keywords: Lie systems; Vessiot-Guldberg-Lie algebra; superposition rule; SODE Lie systems Lie systems; Vessiot-Guldberg-Lie algebra; superposition rule; SODE Lie systems
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Campoamor-Stursberg, R. Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations. Symmetry 2016, 8, 15.

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