{"title":"另一个台球问题","authors":"S. Bolotin, D. Treschev","doi":"10.1134/s106192084010047","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Let <span>\\((M,g)\\)</span> be a Riemannian manifold, <span>\\(\\Omega\\subset M\\)</span> a domain with boundary <span>\\(\\Gamma\\)</span>, and <span>\\(\\phi\\)</span> a smooth function such that <span>\\(\\phi|_\\Omega > 0\\)</span>, <span>\\( \\varphi |_\\Gamma = 0\\)</span>, and <span>\\(d\\phi|_\\Gamma\\ne 0\\)</span>. We study the geodesic flow of the metric <span>\\(G=g/\\phi\\)</span>. The <span>\\(G\\)</span>-distance from any point of <span>\\(\\Omega\\)</span> to <span>\\(\\Gamma\\)</span> is finite, hence, the geodesic flow is incomplete. Regularization of the flow in a neighborhood of <span>\\(\\Gamma\\)</span> establishes a natural reflection law from <span>\\(\\Gamma\\)</span>. This leads to a certain (not quite standard) billiard problem in <span>\\(\\Omega\\)</span>. </p><p> <b> DOI</b> 10.1134/S106192084010047 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Another Billiard Problem\",\"authors\":\"S. Bolotin, D. Treschev\",\"doi\":\"10.1134/s106192084010047\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3 data-test=\\\"abstract-sub-heading\\\">Abstract</h3><p> Let <span>\\\\((M,g)\\\\)</span> be a Riemannian manifold, <span>\\\\(\\\\Omega\\\\subset M\\\\)</span> a domain with boundary <span>\\\\(\\\\Gamma\\\\)</span>, and <span>\\\\(\\\\phi\\\\)</span> a smooth function such that <span>\\\\(\\\\phi|_\\\\Omega > 0\\\\)</span>, <span>\\\\( \\\\varphi |_\\\\Gamma = 0\\\\)</span>, and <span>\\\\(d\\\\phi|_\\\\Gamma\\\\ne 0\\\\)</span>. We study the geodesic flow of the metric <span>\\\\(G=g/\\\\phi\\\\)</span>. The <span>\\\\(G\\\\)</span>-distance from any point of <span>\\\\(\\\\Omega\\\\)</span> to <span>\\\\(\\\\Gamma\\\\)</span> is finite, hence, the geodesic flow is incomplete. Regularization of the flow in a neighborhood of <span>\\\\(\\\\Gamma\\\\)</span> establishes a natural reflection law from <span>\\\\(\\\\Gamma\\\\)</span>. This leads to a certain (not quite standard) billiard problem in <span>\\\\(\\\\Omega\\\\)</span>. </p><p> <b> DOI</b> 10.1134/S106192084010047 </p>\",\"PeriodicalId\":763,\"journal\":{\"name\":\"Russian Journal of Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2024-03-19\",\"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://doi.org/10.1134/s106192084010047\",\"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}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1134/s106192084010047","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}
引用次数: 0
摘要
Abstract Let \((M,g)\) be a Riemannian manifold, \(\Omega\subset M\) a domain with boundary \(\Gamma\), and\(\phi\) a smooth function such that \(\phi|_\Omega > 0\),\( \varphi |_\Gamma = 0\), and\(d\phi|_\Gamma\ne 0\).我们研究度量 \(G=g/\phi\) 的大地流。从\(\Omega\)的任何一点到\(\Gamma\)的距离都是有限的，因此，大地流是不完整的。在（\ω\）的邻域内流动的正则化建立了从（\ω\）到（\ω\）的自然反射定律。这引出了某个（不太标准的）台球问题。 doi 10.1134/s106192084010047
Let \((M,g)\) be a Riemannian manifold, \(\Omega\subset M\) a domain with boundary \(\Gamma\), and \(\phi\) a smooth function such that \(\phi|_\Omega > 0\), \( \varphi |_\Gamma = 0\), and \(d\phi|_\Gamma\ne 0\). We study the geodesic flow of the metric \(G=g/\phi\). The \(G\)-distance from any point of \(\Omega\) to \(\Gamma\) is finite, hence, the geodesic flow is incomplete. Regularization of the flow in a neighborhood of \(\Gamma\) establishes a natural reflection law from \(\Gamma\). This leads to a certain (not quite standard) billiard problem in \(\Omega\).
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.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
