{"title":"具有奇异基础几何结构的动力系统的同步局部正态形式","authors":"Kai Jiang, Tudor S Ratiu and Nguyen Tien Zung","doi":"10.1088/1361-6544/ad700d","DOIUrl":null,"url":null,"abstract":"The aim of this paper is to develop, for the first time, a general theory of simultaneous local normalisation of couples , where X is a dynamical system (vector field) and is an underlying geometric structure preserved by X, even if both have singularities. Such couples appear naturally in many problems, e.g. Hamiltonian dynamics, where is a symplectic structure and one has the theory of Birkhoff normal forms, or constrained dynamics, where is a smooth, in general singular, distribution of tangent subspaces, etc. In this paper, the geometric structure is of the following types: volume form, symplectic form, contact form, Poisson tensor, as well as their singular versions. The paper addresses mainly the more difficult situations when both X and are singular at a point and its results prove the existence of natural simultaneous normal forms in these cases. In general, the normalisation is only formal, but when and X are (real or complex) analytic and X is analytically or Darboux integrable, then the simultaneous normalisation is also analytic. Our theory is based on a new approach, called the Toric Conservation Principle, as well as the classical step-by-step normalisation technique, and the equivariant path method.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"15 1","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Simultaneous local normal forms of dynamical systems with singular underlying geometric structures\",\"authors\":\"Kai Jiang, Tudor S Ratiu and Nguyen Tien Zung\",\"doi\":\"10.1088/1361-6544/ad700d\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The aim of this paper is to develop, for the first time, a general theory of simultaneous local normalisation of couples , where X is a dynamical system (vector field) and is an underlying geometric structure preserved by X, even if both have singularities. Such couples appear naturally in many problems, e.g. Hamiltonian dynamics, where is a symplectic structure and one has the theory of Birkhoff normal forms, or constrained dynamics, where is a smooth, in general singular, distribution of tangent subspaces, etc. In this paper, the geometric structure is of the following types: volume form, symplectic form, contact form, Poisson tensor, as well as their singular versions. The paper addresses mainly the more difficult situations when both X and are singular at a point and its results prove the existence of natural simultaneous normal forms in these cases. In general, the normalisation is only formal, but when and X are (real or complex) analytic and X is analytically or Darboux integrable, then the simultaneous normalisation is also analytic. Our theory is based on a new approach, called the Toric Conservation Principle, as well as the classical step-by-step normalisation technique, and the equivariant path method.\",\"PeriodicalId\":54715,\"journal\":{\"name\":\"Nonlinearity\",\"volume\":\"15 1\",\"pages\":\"\"},\"PeriodicalIF\":1.6000,\"publicationDate\":\"2024-09-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinearity\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1088/1361-6544/ad700d\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinearity","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1088/1361-6544/ad700d","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
本文的目的是首次提出一种对偶同时局部归一化的一般理论,其中 X 是一个动力学系统(向量场),X 是一个由 X 保留的底层几何结构,即使两者都有奇点。这种耦合自然出现在许多问题中,例如汉密尔顿动力学,其中 X 是交映结构,我们有伯克霍夫正形式理论;或约束动力学,其中 X 是切分子空间的平滑分布,一般是奇异分布,等等。在本文中,几何结构有以下几种类型:体积形式、交映形式、接触形式、泊松张量以及它们的奇异版本。本文主要讨论了当 X 和都是奇异点时较为困难的情况,其结果证明了在这些情况下存在自然的同时正则表达式。一般来说,正化只是形式上的,但当和 X 是(实或复)解析的,并且 X 是解析或达布可积分的,那么同时正化也是解析的。我们的理论基于一种新方法,即 "环守恒原理",以及经典的分步归一化技术和等变路径法。
Simultaneous local normal forms of dynamical systems with singular underlying geometric structures
The aim of this paper is to develop, for the first time, a general theory of simultaneous local normalisation of couples , where X is a dynamical system (vector field) and is an underlying geometric structure preserved by X, even if both have singularities. Such couples appear naturally in many problems, e.g. Hamiltonian dynamics, where is a symplectic structure and one has the theory of Birkhoff normal forms, or constrained dynamics, where is a smooth, in general singular, distribution of tangent subspaces, etc. In this paper, the geometric structure is of the following types: volume form, symplectic form, contact form, Poisson tensor, as well as their singular versions. The paper addresses mainly the more difficult situations when both X and are singular at a point and its results prove the existence of natural simultaneous normal forms in these cases. In general, the normalisation is only formal, but when and X are (real or complex) analytic and X is analytically or Darboux integrable, then the simultaneous normalisation is also analytic. Our theory is based on a new approach, called the Toric Conservation Principle, as well as the classical step-by-step normalisation technique, and the equivariant path method.
期刊介绍:
Aimed primarily at mathematicians and physicists interested in research on nonlinear phenomena, the journal''s coverage ranges from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant physical and mathematical interest.
Subject coverage:
The journal publishes papers on nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance. A more detailed indication is given by the subject interests of the Editorial Board members, which are listed in every issue of the journal.
Due to the broad scope of Nonlinearity, and in order to make all papers published in the journal accessible to its wide readership, authors are required to provide sufficient introductory material in their paper. This material should contain enough detail and background information to place their research into context and to make it understandable to scientists working on nonlinear phenomena.
Nonlinearity is a journal of the Institute of Physics and the London Mathematical Society.