{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:54:17Z","timestamp":1760234057688,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,17]],"date-time":"2021-03-17T00:00:00Z","timestamp":1615939200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PID2019-106820RBC22"],"award-info":[{"award-number":["PID2019-106820RBC22"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100014180","name":"Junta de Castilla y Le\u00f3n","doi-asserted-by":"publisher","award":["SA256P18"],"award-info":[{"award-number":["SA256P18"]}],"id":[{"id":"10.13039\/501100014180","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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 \u201cone dimensional\u201d 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.<\/jats:p>","DOI":"10.3390\/sym13030493","type":"journal-article","created":{"date-parts":[[2021,3,17]],"date-time":"2021-03-17T21:43:31Z","timestamp":1616017411000},"page":"493","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Review in Ermakov Systems and Their Symmetries"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1880-4723","authenticated-orcid":false,"given":"Jose M.","family":"Cerver\u00f3","sequence":"first","affiliation":[{"name":"Department of Fundamental Physics, Universidad de Salamanca, 37008 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9026-3979","authenticated-orcid":false,"given":"Pilar G.","family":"Est\u00e9vez","sequence":"additional","affiliation":[{"name":"Department of Fundamental Physics, Universidad de Salamanca, 37008 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,17]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Second order differential equations. 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