Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations
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 equatio...
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| Format: | Article |
| Language: | English |
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MDPI AG
2016-03-01
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| Series: | Symmetry |
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| Online Access: | http://www.mdpi.com/2073-8994/8/3/15 |
| _version_ | 1798024331345788928 |
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| author | Rutwig Campoamor-Stursberg |
| author_facet | Rutwig Campoamor-Stursberg |
| author_sort | Rutwig Campoamor-Stursberg |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T18:00:29Z |
| format | Article |
| id | doaj.art-0a175abe2d6247b0ab5ed497bc32ec2d |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T18:00:29Z |
| publishDate | 2016-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-0a175abe2d6247b0ab5ed497bc32ec2d2022-12-22T04:10:28ZengMDPI AGSymmetry2073-89942016-03-01831510.3390/sym8030015sym8030015Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential EquationsRutwig Campoamor-Stursberg0Instituto de Matemática Interdisciplinar and Depto. Geometría y Topología, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid E-28040, SpainA 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.http://www.mdpi.com/2073-8994/8/3/15Lie systemsVessiot-Guldberg-Lie algebrasuperposition ruleSODE Lie systems |
| spellingShingle | Rutwig Campoamor-Stursberg Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations Symmetry Lie systems Vessiot-Guldberg-Lie algebra superposition rule SODE Lie systems |
| title | Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations |
| title_full | Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations |
| title_fullStr | Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations |
| title_full_unstemmed | Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations |
| title_short | Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations |
| title_sort | low dimensional vessiot guldberg lie algebras of second order ordinary differential equations |
| topic | Lie systems Vessiot-Guldberg-Lie algebra superposition rule SODE Lie systems |
| url | http://www.mdpi.com/2073-8994/8/3/15 |
| work_keys_str_mv | AT rutwigcampoamorstursberg lowdimensionalvessiotguldbergliealgebrasofsecondorderordinarydifferentialequations |