北京雁栖湖应用数学研究院 北京雁栖湖应用数学研究院

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关于我们
院长致辞
理事会
协作机构
参观来访
人员
管理层
科研人员
博士后
来访学者
行政团队
学术研究
研究团队
公开课
讨论班
招生招聘
教研人员
博士后
学生
会议
学术会议
工作坊
论坛
学院生活
住宿
交通
配套设施
周边旅游
新闻
新闻动态
通知公告
资料下载
清华大学 "求真书院"
清华大学丘成桐数学科学中心
清华三亚国际数学论坛
上海数学与交叉学科研究院
BIMSA > Introduction to Geometries and Mechanics \(ICBS\)
Introduction to Geometries and Mechanics
In classical mechanics, we study the evolution of a given system in configuration space. The properties of this space are described by Euclidean, projective, Riemannian, pseudo and sub-Riemannian geometries. In Hamiltonian mechanics, we have a phase space and use symplectic, Poisson, and contact geometries. To solve the equations of motion, we also use algebraic, differential, and so on other geometries. The most important and interesting mathematical models lie at the intersection of these geometric theories.

In this course we will start with Euclidean geometry and discuss general concepts such as metrics, geodesic motion, isometries, symmetries, Noether's theorem, Killing vector fields, hidden symmetries, equivalent metrics, Killing tensors, flatness and reduction of bilinear forms, orthogonal curvilinear coordinates, and so on. Then we consider some examples of constrained Hamiltonian mechanics imposing holonomic and nonholomic constraints. We then discuss the Erlangen program proposed by Klein, which is a method of characterizing geometries based on group theory and projective geometry. Some applications of Riemannian and pseudo-Riemannian geometry in mechanics have also been considered. Finally, the role of sub-Riemannian geometry in control theory is briefly discussed.

The main aim of this course is to give an overview of modern geometric methods used in various branches of classical mechanics to solve open problems. Students can discuss possible approaches to solving these open problems separately with the instructor.
讲师
安德烈·茨加诺夫
日期
2023年10月10日 至 12月28日
位置
Weekday Time Venue Online ID Password
周二,周四 09:50 - 11:25 A3-4-101 ZOOM 3 361 038 6975 BIMSA
修课要求
Knowledge of the basic concepts of classical mechanics is required. Knowledge of representation theory, metric space theory, and tensor analysis would be useful.
参考资料
The material is self-sufficient but basic definitions and their discussions may be found in:
1. Agrachev A.A., Sachkov Yu.L. (2004), Control Theory from the Geometric Viewpoint.
2. Aldrovandi R., Pereira J.C. (2017), An Introduction to Geometrical Physics.
3. Bloch A.M. (2003) Nonholonomic Mechanics and Control.
4. Coxeter H. S. M. (1961), Introduction to Geometry.
5. Frolov V.P., Zelnikov A., (2015), Introduction to Black Hole Physics.
6. Hamel G. (1949), Teoretische Mechanik.
7. Klein F. (2004), Elementary Mathematics from an Advanced Standpoint: Geometry.
8. Routh E. (1884), Advanced part of a Treatise on the Dynamics of a System of Rigid Bodies.
9. Schouten J.A. (1954), Ricci-Calculus: An Introduction to Tensor Analysis and Its Geometrical Applications.
听众
Graduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Andrey Tsiganov currently works at the Department of Computational Physics, Saint Petersburg State University, Russia. His main research interests are integrable and superintegrable systems in classical and quantum mechanics, nonholonomic and vakonomic mechanics, geometry and topology of dynamical systems, see profile at https://www.researchgate.net/profile/Andrey-Tsiganov. He is one of the organizers of the BIMSA Integrable System Seminar, see https://researchseminars.org/seminar/BIMSA-ISS and https://sites.google.com/view/bimsa-iss.
北京雁栖湖应用数学研究院
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北京市怀柔区 河防口村544号
北京雁栖湖应用数学研究院 101408

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Email. administration@bimsa.cn

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