Group-theoretical methods in classical integrable systems
This course introduces group-theoretical methods in integrable systems and their geometric foundations. A central role is played by Lie group actions and Hamiltonian reduction (the Marsden–Weinstein–Meyer theorem), illustrated by the derivation of the Calogero–Moser system from free motion. The course also introduces the classical r-matrix formalism, the classical Yang–Baxter equation, and integrable models related to semisimple Lie algebras, including Toda chains and the Gaudin model. Finally, we discuss integrable nonlinear equations such as the KdV hierarchy and their description via Drinfeld–Sokolov reduction and Lax pairs.
讲师
日期
2026年03月11日 至 05月28日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周三 | 13:30 - 16:55 | A3-1a-204 | Zoom Ivan Sechin | 897 4738 4798 | 1 |
修课要求
Basic knowledge of differential geometry, Lie theory, classical and quantum mechanics.
课程大纲
1. Introduction. Examples of integrable systems. Solution by quadrature, Liouville’s theorem, action–angle variables. Lax pair, construction of integrals of motion. Calogero–Moser system of particles, its Lax representation.
[BBT] Chapters 2, 7
[Arn] Chapter 10
[Per] Paragraphs 2.1–3.2
[Aru] Chapter 1.1
[BBT] Chapters 2, 7
[Arn] Chapter 10
[Per] Paragraphs 2.1–3.2
[Aru] Chapter 1.1
参考资料
[BBT] O. Babelon, D. Bernard, M. Talon. Introduction to classical integrable systems. Cambridge University Press, 2003.
[Arn] V. I. Arnold. Mathematical methods of classical mechanics. Springer New York, 1978.
[Per] A. M. Perelomov. Integrable Systems of Classical Mechanics and Lie Algebras. Springer Basel AG, 1989.
[Aru] G. Arutyunov. Elements of Classical and Quantum Integrable Systems. Springer Nature Switzerland AG, 2019.
[Sil] A. C. da Silva. Lectures on Symplectic Geometry. https://people.math.ethz.ch/~acannas/Papers/lsg.pdf
[AM] A. Alekseev, I. Marshall. Lecture course: Introduction to Symplectic Geometry, Moment Maps, Localization and Integrability. https://sites.google.com/view/alexeevsymplectic22
[MJD] T. Miwa, M. Jimbo, E. Date. Solitons: Differential equations, symmetries, and infinite-dimensional algebras. Cambridge University Press, 2000.
[Arn] V. I. Arnold. Mathematical methods of classical mechanics. Springer New York, 1978.
[Per] A. M. Perelomov. Integrable Systems of Classical Mechanics and Lie Algebras. Springer Basel AG, 1989.
[Aru] G. Arutyunov. Elements of Classical and Quantum Integrable Systems. Springer Nature Switzerland AG, 2019.
[Sil] A. C. da Silva. Lectures on Symplectic Geometry. https://people.math.ethz.ch/~acannas/Papers/lsg.pdf
[AM] A. Alekseev, I. Marshall. Lecture course: Introduction to Symplectic Geometry, Moment Maps, Localization and Integrability. https://sites.google.com/view/alexeevsymplectic22
[MJD] T. Miwa, M. Jimbo, E. Date. Solitons: Differential equations, symmetries, and infinite-dimensional algebras. Cambridge University Press, 2000.
听众
Undergraduate
, Advanced Undergraduate
, Graduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Ivan Sechin has defended a PhD thesis on Mathematical Physics at Skoltech in 2022 and joined BIMSA in 2022. His research interests are mainly devoted to classical and quantum integrable systems.