{"title":"Classification of Singularities of the Liouville Foliation of an Integrable Elliptical Billiard with a Potential of Fourth Degree","authors":"S.E. Pustovoitov","doi":"10.1134/S1061920823040155","DOIUrl":null,"url":null,"abstract":"<p> The paper is devoted to the study of a billiard bounded by an ellipse and equipped with a fourth degree potential as an integrable Hamiltonian system with two degrees of freedom. In previous works, the author described the structure of the Liouville foliation of such a system on nonsingular levels of the Hamiltonian in terms of Fomenko–Zieschang invariants: marked molecules and 3-atoms. Moreover, the dependence of the structure of the bifurcation diagram on the parameters of the potential has been established. The present work continues this study. Thus, the structure of the Liouville foliation in a neighborhood of critical layers containing a nondegenerate singular point of rank 0 or a degenerate orbit has been described. A classification of the obtained semilocal singularities was given. Finally, connections of our system with well-known cases of rigid body dynamics containing equivalent singularities is established. </p><p> <b> DOI</b> 10.1134/S1061920823040155 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"643 - 673"},"PeriodicalIF":1.7000,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1134/S1061920823040155","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
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Abstract
The paper is devoted to the study of a billiard bounded by an ellipse and equipped with a fourth degree potential as an integrable Hamiltonian system with two degrees of freedom. In previous works, the author described the structure of the Liouville foliation of such a system on nonsingular levels of the Hamiltonian in terms of Fomenko–Zieschang invariants: marked molecules and 3-atoms. Moreover, the dependence of the structure of the bifurcation diagram on the parameters of the potential has been established. The present work continues this study. Thus, the structure of the Liouville foliation in a neighborhood of critical layers containing a nondegenerate singular point of rank 0 or a degenerate orbit has been described. A classification of the obtained semilocal singularities was given. Finally, connections of our system with well-known cases of rigid body dynamics containing equivalent singularities is established.
期刊介绍：
Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.
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