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.
Lecturer
Date
11th March ~ 28th May, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday | 13:30 - 16:55 | A3-1a-204 | Zoom Ivan Sechin | 897 4738 4798 | 1 |
Prerequisite
Basic knowledge of differential geometry, Lie theory, classical and quantum mechanics.
Syllabus
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
Reference
[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.
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
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.