{"title":"A Top on a Vibrating Base: New Integrable Problem of Nonholonomic Mechanics","authors":"Alexey V. Borisov, Alexander P. Ivanov","doi":"10.1134/S1560354722010026","DOIUrl":null,"url":null,"abstract":"<div><p>A spherical rigid body rolling without sliding on a horizontal support is considered. The body is axially symmetric but unbalanced (tippe top). The support performs high-frequency oscillations with small amplitude. To implement the standard averaging procedure, we present equations of motion in quasi-coordinates in Hamiltonian form with additional terms of nonholonomicity [16] and introduce a new fast time variable. The averaged system is similar to the initial one with an additional term, known as vibrational potential [8, 9, 18]. This term depends on the single variable — the nutation angle <span>\\(\\theta\\)</span>, and according to the work of Chaplygin [5], the averaged system is integrable. Some examples exhibit the influence of vibrations on the dynamics.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"27 1","pages":"2 - 10"},"PeriodicalIF":0.8000,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Regular and Chaotic Dynamics","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1134/S1560354722010026","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: 2
Abstract
A spherical rigid body rolling without sliding on a horizontal support is considered. The body is axially symmetric but unbalanced (tippe top). The support performs high-frequency oscillations with small amplitude. To implement the standard averaging procedure, we present equations of motion in quasi-coordinates in Hamiltonian form with additional terms of nonholonomicity [16] and introduce a new fast time variable. The averaged system is similar to the initial one with an additional term, known as vibrational potential [8, 9, 18]. This term depends on the single variable — the nutation angle \(\theta\), and according to the work of Chaplygin [5], the averaged system is integrable. Some examples exhibit the influence of vibrations on the dynamics.
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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