A Review in Ermakov Systems and Their Symmetries
A review of the mathematical and physical aspects of the Ermakov systems is presented. The main properties of Lie algebra invariants are extensively used. The two and tridimensional Ermakov systems are fully analyzed and the correspondent invariants found. Then, we go over quantization with special...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-03-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/13/3/493 |
| _version_ | 1797540983200546816 |
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| author | Jose M. Cerveró Pilar G. Estévez |
| author_facet | Jose M. Cerveró Pilar G. Estévez |
| author_sort | Jose M. Cerveró |
| collection | DOAJ |
| description | A review of the mathematical and physical aspects of the Ermakov systems is presented. The main properties of Lie algebra invariants are extensively used. The two and tridimensional Ermakov systems are fully analyzed and the correspondent invariants found. Then, we go over quantization with special emphasis in the two dimensional case. An application to Nonlinear Optics is hereby developed. We also treat the so-called “one dimensional” case, which is easily solved in the classical case but offers special interest in the quantum realm, where one can find exactly the Feynman propagator. We finish with the stationary phase approximation which contains also some interesting features when compared with the exact solution. Some prospects for future research are also discussed. |
| first_indexed | 2024-03-10T13:08:47Z |
| format | Article |
| id | doaj.art-8814437fee074e5ea81019fce2e9cc8e |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T13:08:47Z |
| publishDate | 2021-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-8814437fee074e5ea81019fce2e9cc8e2023-11-21T10:55:13ZengMDPI AGSymmetry2073-89942021-03-0113349310.3390/sym13030493A Review in Ermakov Systems and Their SymmetriesJose M. Cerveró0Pilar G. Estévez1Department of Fundamental Physics, Universidad de Salamanca, 37008 Salamanca, SpainDepartment of Fundamental Physics, Universidad de Salamanca, 37008 Salamanca, SpainA review of the mathematical and physical aspects of the Ermakov systems is presented. The main properties of Lie algebra invariants are extensively used. The two and tridimensional Ermakov systems are fully analyzed and the correspondent invariants found. Then, we go over quantization with special emphasis in the two dimensional case. An application to Nonlinear Optics is hereby developed. We also treat the so-called “one dimensional” case, which is easily solved in the classical case but offers special interest in the quantum realm, where one can find exactly the Feynman propagator. We finish with the stationary phase approximation which contains also some interesting features when compared with the exact solution. Some prospects for future research are also discussed.https://www.mdpi.com/2073-8994/13/3/493ermakov systemsymmetry of dynamical systemintegrability |
| spellingShingle | Jose M. Cerveró Pilar G. Estévez A Review in Ermakov Systems and Their Symmetries Symmetry ermakov system symmetry of dynamical system integrability |
| title | A Review in Ermakov Systems and Their Symmetries |
| title_full | A Review in Ermakov Systems and Their Symmetries |
| title_fullStr | A Review in Ermakov Systems and Their Symmetries |
| title_full_unstemmed | A Review in Ermakov Systems and Their Symmetries |
| title_short | A Review in Ermakov Systems and Their Symmetries |
| title_sort | review in ermakov systems and their symmetries |
| topic | ermakov system symmetry of dynamical system integrability |
| url | https://www.mdpi.com/2073-8994/13/3/493 |
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