Algebraic Structures in Integrable Systems
https://itmp.msu.ru/en/mscgeom...
The theory of integrable systems is a modern field of scientific research at the intersection of dynamical systems theory, discrete mathematics, quantum physics, low-dimensional topology and statistical mechanics.
The course is a first introduction to the concept of integrability. This semester is devoted to the concept of Liouville integrability, some algebraic and geometric constructions underlying the phenomenon of integrability, including: the Lax representation, the Bihamiltonian structure, r-matrix Poisson structures and Lie-Poisson groups, the Adler-Kostant-Symes scheme.
The course is accompanied by an intensive analysis of examples and applied problems reflecting the property of integrability. The main examples are the harmonic oscillator, the Kepler problem, the Euler top, the open Toda chain, the full symmetric Toda system, the KdV equation, the nonlinear Schrodinger equation.
At the beginning of the course, we will pay attention to the basic elements of the theory of Lie groups and Lie algebras and differential geometry, which are necessary to understand the main part of the course.
- 01:05:29Lecture 1. Basic Notions, Part 1
- 01:25:35Lecture 2. Basic Notions, Part 2
- 01:25:46Lecture 3. Poisson Structure
- 01:21:23Lecture 4. Basic Notions: Exercises. Casimir Functions
- 01:19:15Lecture 5. Symplectic Structure. Hamiltonian Vector Fields
- 01:14:28Lecture 6. Poisson Center: Exercises
- 01:27:21Lecture 7. Examples of Hamiltonian Systems. Liouville-Arnold Integrability
- 01:05:43Lecture 8. Hamiltonian Vector Fields: Exercises
- 01:22:48Lecture 9. The Liouville-Arnold Theorem. KAM Theorem. Lax Representation
- 01:19:08Lecture 10. Liouville Integrability: Exercises
- 01:18:35Lecture 11. Integrals of Motion for the Open Toda Chain
- 01:01:40Lecture 12. Flaschka Сoordinates: Exercises
- 01:35:12Lecture 13. Open Toda Chain: R-Matrices
- 53:29Lecture 14. R-Matrices: Exercises
- 01:22:07Lecture 15. Open Toda Chain: QR-Decomposition
- 01:16:33Lecture 16. R-Matrices: Exercises. Full Symmetric Toda System
- 01:22:31Lecture 17. AKS-Method. Full Symmetric Toda System
- 01:16:12Lecture 18. Full Symmetric Toda System: Exersises
- 01:19:41Lecture 19. AKS Scheme and the Full Symmetric Toda System
- 01:04:03Lecture 20. AKS-Scheme: Exersises
- 01:17:52Lecture 21. R-Matrix Structures Related With the Full Symmetric Toda System. Gaudin Model
- 48:42Lecture 22. Full Symmetric Toda System: Exercises. Spectral Curve
- 01:17:34Lecture 23. Gaudin Model. Separation of Variables
- 01:13:17Lecture 24. R-Matrix Structures Related With the Full Symmetric Toda System: Exercises. Stachel Systeme
- 01:14:51Lecture 25. Bihamiltonian Formalism. Argument Shift Method
- 01:07:10Lecture 26. Argument Shift Method. Stachel Systeme: Exersises
- 01:24:04Lecture 27. Bihamiltonian Formalism. Lenard-Magri Scheme. Calogero-Moser System
- 01:10:12Lecture 28. Schouten Bracket: Exercises
- 01:12:57Lecture 29. Kadomtsev-Petviashvili Equation: Linear Problems. KdV Equation
- 01:03:29Lecture 30. KdV Equation: Theory and Exercises