{"title":"Solvable Algebras and Integrable Systems","authors":"Valery V. Kozlov","doi":"10.1134/S1560354724520022","DOIUrl":null,"url":null,"abstract":"<div><p>This paper discusses a range of questions concerning the application of\nsolvable Lie algebras of vector fields to exact integration of systems of ordinary\ndifferential equations. The set of <span>\\(n\\)</span> independent vector fields\ngenerating a solvable Lie algebra in <span>\\(n\\)</span>-dimensional space is locally\nreduced to some “canonical” form. This reduction is performed constructively (using\nquadratures), which, in particular, allows a simultaneous integration of <span>\\(n\\)</span> systems of\ndifferential equations that are generated by these fields.\nGeneralized completely integrable systems are introduced and their properties are investigated.\nGeneral ideas are applied to integration of the Hamiltonian systems of differential equations.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 5","pages":"717 - 727"},"PeriodicalIF":0.8000,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Regular and Chaotic Dynamics","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1134/S1560354724520022","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This paper discusses a range of questions concerning the application of
solvable Lie algebras of vector fields to exact integration of systems of ordinary
differential equations. The set of \(n\) independent vector fields
generating a solvable Lie algebra in \(n\)-dimensional space is locally
reduced to some “canonical” form. This reduction is performed constructively (using
quadratures), which, in particular, allows a simultaneous integration of \(n\) systems of
differential equations that are generated by these fields.
Generalized completely integrable systems are introduced and their properties are investigated.
General ideas are applied to integration of the Hamiltonian systems of differential equations.
期刊介绍：
Regular and Chaotic Dynamics (RCD) is an international journal publishing original research papers in dynamical systems theory and its applications. Rooted in the Moscow school of mathematics and mechanics, the journal successfully combines classical problems, modern mathematical techniques and breakthroughs in the field. Regular and Chaotic Dynamics welcomes papers that establish original results, characterized by rigorous mathematical settings and proofs, and that also address practical problems. In addition to research papers, the journal publishes review articles, historical and polemical essays, and translations of works by influential scientists of past centuries, previously unavailable in English. Along with regular issues, RCD also publishes special issues devoted to particular topics and events in the world of dynamical systems.
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