{"title":"雅可比最后乘法器和二维超可整定振荡器","authors":"Akash Sinha, Aritra Ghosh","doi":"10.1007/s12043-024-02786-3","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we examine the role of the Jacobi last multiplier in the context of two-dimensional oscillators. We first consider two-dimensional unit-mass oscillators admitting a separable Hamiltonian description, i.e., <span>\\(H = H_1 + H_2\\)</span>, where <span>\\(H_1\\)</span> and <span>\\(H_2\\)</span> are the Hamiltonians of two one-dimensional unit-mass oscillators. It is shown that there exists a third functionally-independent first integral <span>\\(\\Theta \\)</span>, ensuring superintegrability. Various examples are explicitly worked out. We then consider position-dependent-mass oscillators and the Bateman pair, where the latter consists of a pair of dissipative linear oscillators. Quite remarkably, the Bateman pair is found to be superintegrable, despite admitting a Hamiltonian which cannot be separated into two isolated (non-interacting) one-dimensional oscillators.\n</p></div>","PeriodicalId":743,"journal":{"name":"Pramana","volume":"98 3","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Jacobi last multiplier and two-dimensional superintegrable oscillators\",\"authors\":\"Akash Sinha, Aritra Ghosh\",\"doi\":\"10.1007/s12043-024-02786-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we examine the role of the Jacobi last multiplier in the context of two-dimensional oscillators. We first consider two-dimensional unit-mass oscillators admitting a separable Hamiltonian description, i.e., <span>\\\\(H = H_1 + H_2\\\\)</span>, where <span>\\\\(H_1\\\\)</span> and <span>\\\\(H_2\\\\)</span> are the Hamiltonians of two one-dimensional unit-mass oscillators. It is shown that there exists a third functionally-independent first integral <span>\\\\(\\\\Theta \\\\)</span>, ensuring superintegrability. Various examples are explicitly worked out. We then consider position-dependent-mass oscillators and the Bateman pair, where the latter consists of a pair of dissipative linear oscillators. Quite remarkably, the Bateman pair is found to be superintegrable, despite admitting a Hamiltonian which cannot be separated into two isolated (non-interacting) one-dimensional oscillators.\\n</p></div>\",\"PeriodicalId\":743,\"journal\":{\"name\":\"Pramana\",\"volume\":\"98 3\",\"pages\":\"\"},\"PeriodicalIF\":1.9000,\"publicationDate\":\"2024-07-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Pramana\",\"FirstCategoryId\":\"4\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s12043-024-02786-3\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Pramana","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1007/s12043-024-02786-3","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
Jacobi last multiplier and two-dimensional superintegrable oscillators
In this paper, we examine the role of the Jacobi last multiplier in the context of two-dimensional oscillators. We first consider two-dimensional unit-mass oscillators admitting a separable Hamiltonian description, i.e., \(H = H_1 + H_2\), where \(H_1\) and \(H_2\) are the Hamiltonians of two one-dimensional unit-mass oscillators. It is shown that there exists a third functionally-independent first integral \(\Theta \), ensuring superintegrability. Various examples are explicitly worked out. We then consider position-dependent-mass oscillators and the Bateman pair, where the latter consists of a pair of dissipative linear oscillators. Quite remarkably, the Bateman pair is found to be superintegrable, despite admitting a Hamiltonian which cannot be separated into two isolated (non-interacting) one-dimensional oscillators.
期刊介绍:
Pramana - Journal of Physics is a monthly research journal in English published by the Indian Academy of Sciences in collaboration with Indian National Science Academy and Indian Physics Association. The journal publishes refereed papers covering current research in Physics, both original contributions - research papers, brief reports or rapid communications - and invited reviews. Pramana also publishes special issues devoted to advances in specific areas of Physics and proceedings of select high quality conferences.