{"title":"寻找具有表面张力的水波问题的对称性","authors":"E. Shamardina","doi":"10.5922/0321-4796-2022-53-13","DOIUrl":null,"url":null,"abstract":"T. Brooke Benjamin and P. J. Olver “Hamiltonian structure, symmetries and conservation laws for water waves” study the behavior of Hamiltonian systems with an infinite-dimensional phase space. The methods of variational problems and infinite-dimensional differential geometry are applicable to this problem. A special case of the problem is an abstract problem of hydrodynamics for an ideal fluid. Its configuration space is the group of volume-preserving diffeomorphisms of some manifold in or filled with fluid. Even more special is the problem of waves on water. Its non-standard nature is due to the presence of boundary conditions on the free surface. These boundary conditions can be interpreted in terms of the functional derivatives of the energy integral, which plays the role of the Hamiltonian. Here we consider in detail the case of this problem in R2, taking into account surface tension, and find symmetries for it, which was not considered in detail in the article. Finding symmetries can be achieved without recourse to the Hamiltonian structure of the given problem.","PeriodicalId":114406,"journal":{"name":"Differential Geometry of Manifolds of Figures","volume":"83 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Finding symmetries for the problem of water waves with surface tension\",\"authors\":\"E. Shamardina\",\"doi\":\"10.5922/0321-4796-2022-53-13\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"T. Brooke Benjamin and P. J. Olver “Hamiltonian structure, symmetries and conservation laws for water waves” study the behavior of Hamiltonian systems with an infinite-dimensional phase space. The methods of variational problems and infinite-dimensional differential geometry are applicable to this problem. A special case of the problem is an abstract problem of hydrodynamics for an ideal fluid. Its configuration space is the group of volume-preserving diffeomorphisms of some manifold in or filled with fluid. Even more special is the problem of waves on water. Its non-standard nature is due to the presence of boundary conditions on the free surface. These boundary conditions can be interpreted in terms of the functional derivatives of the energy integral, which plays the role of the Hamiltonian. Here we consider in detail the case of this problem in R2, taking into account surface tension, and find symmetries for it, which was not considered in detail in the article. Finding symmetries can be achieved without recourse to the Hamiltonian structure of the given problem.\",\"PeriodicalId\":114406,\"journal\":{\"name\":\"Differential Geometry of Manifolds of Figures\",\"volume\":\"83 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1900-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Differential Geometry of Manifolds of Figures\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.5922/0321-4796-2022-53-13\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Geometry of Manifolds of Figures","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5922/0321-4796-2022-53-13","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
T. Brooke Benjamin和P. J. Olver“水波的哈密顿结构、对称性和守恒定律”研究了具有无限维相空间的哈密尔-年系统的行为。变分问题的方法和无穷维微分几何的方法适用于这个问题。该问题的一个特例是理想流体的抽象流体力学问题。它的位形空间是在流体中或充满流体的流形的保体积微分同态的群。更特殊的是水上波浪的问题。它的非标准性质是由于在自由表面上存在边界条件。这些边界条件可以用能量积分的泛函导数来解释,它起着哈密顿量的作用。在这里,我们详细考虑了R2中这个问题的情况,考虑了表面张力,并找到了它的对称性,这在文章中没有详细考虑。不借助给定问题的哈密顿结构也能找到对称性。
Finding symmetries for the problem of water waves with surface tension
T. Brooke Benjamin and P. J. Olver “Hamiltonian structure, symmetries and conservation laws for water waves” study the behavior of Hamiltonian systems with an infinite-dimensional phase space. The methods of variational problems and infinite-dimensional differential geometry are applicable to this problem. A special case of the problem is an abstract problem of hydrodynamics for an ideal fluid. Its configuration space is the group of volume-preserving diffeomorphisms of some manifold in or filled with fluid. Even more special is the problem of waves on water. Its non-standard nature is due to the presence of boundary conditions on the free surface. These boundary conditions can be interpreted in terms of the functional derivatives of the energy integral, which plays the role of the Hamiltonian. Here we consider in detail the case of this problem in R2, taking into account surface tension, and find symmetries for it, which was not considered in detail in the article. Finding symmetries can be achieved without recourse to the Hamiltonian structure of the given problem.