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.
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. 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
2. Basics of symplectic geometry. Symplectic linear algebra. Symplectic manifolds. Darboux theorem. Poisson manifolds and symplectic leaves. Symplectic structure on a cotangent bundle of a manifold: Liouville 1-form and canonical symplectic form. Lagrangian submanifolds. Weinstein Lagrangian neighborhood theorem.
[Sil] Chapters 1–2, 18
[BBT] Chapter 14
[AM] Lectures 1–3
[Per] Paragraph 1.3
3. Vector fields on smooth manifolds. One-parameter groups of diffeomorphisms. The graded algebra of differential forms. Derivations of differential forms, contraction with a vector field, and Lie derivative. Magic Cartan's formula. Symplectic and Hamiltonian vector fields on symplectic manifolds.
[Sil] Chapter 18
[BBT] Chapter 14
[AM] Lectures 2–3
4. Lie groups. Smooth actions of Lie groups on smooth manifolds. Left-invariant vector fields, tangent Lie algebra. Maurer–Cartan form. Fundamental vector fields. Adjoint and coadjoint actions of Lie groups. Poisson brackets on a dual to a Lie algebra. Coadjoint orbits of Lie groups, Kirillov–Kostant symplectic structure on coadjoint orbits.
[Kir] Chapters 2–3
[Sil] Chapters 21–22
[AM] Lectures 4–5
5. Symplectic, weakly Hamiltonian, and Hamiltonian actions of Lie groups on symplectic manifolds. Momentum map. Chevalley–Eilenberg complex, Lie algebra cohomologies. Cohomological conditions of existence and uniqueness of the momentum map. Relation to central extensions.
[Sil] Chapters 21–22, 26
[AM] Lectures 5–6
[Wei] Chapter 7
6. Properties of the momentum map. Momentum map as a conservation law for an invariant Hamiltonian. Example: motion in a central field, SO(3) symmetry, angular momentum. Momentum maps for natural actions on cotangent bundles and coadjoint orbits. Classification of coadjoint orbits of unitary group U(n): flag manifolds, Grassmannians, complex projective spaces. Fubini–Study form as a symplectic form on CP^{n - 1}.
[Sil] Chapters 21–22, 26
[AM] Lectures 5–6
7. Actions of compact Lie groups on symplectic manifolds. Regular values of the momentum map. Marsden–Weinstein–Meyer theorem. Symplectic reduction, construction of the symplectic form on the symplectic quotient. Projection method for dynamics given by invariant Hamiltonians. Example: motion in a central field, centrifugal potential.
[Sil] Chapters 23–24
[AM] Lectures 7–8
8. Application of the symplectic reduction method to classical integrable systems. Cotangent bundle of a Lie algebra, reduction with respect to the adjoint Lie group action. Example: unitary group U(n), reduction along minimal nontrivial coadjoint orbit, and the Calogero–Moser system of particles. Lax matrix of the Calogero–Moser system from reduction. Solution of Calogero–Moser equations of motion.
[KKS]
[Per]
9. Cotangent bundle of a Lie group, trivialization by left and right shifts. Iwasawa decomposition. Symplectic reduction with respect to the action of Lie subgroups, integrable Toda chains.
10. Classical r-matrix method. Classical Yang–Baxter equation, modified classical Yang–Baxter equation. Involutivity of Hamiltonians from the classical r-matrix structure. Existence of a classical r-matrix. Classical r-matrices for Calogero–Moser and Toda integrable systems.
11. Lax representation with a spectral parameter. Example: integrable tops. Spin Calogero–Moser system. Classical Yang–Baxter equation with spectral parameters. Permutation matrix and corresponding classical r-matrix. Gaudin integrable model.
12. Integrable systems related to semisimple Lie algebras. Review of Lie algebra theory: root systems and classification of semisimple Lie algebras. Toda chains constructed by a root system. Affine Lie algebras and periodic Toda chain.
[BBT] Chapters 2, 7
[Arn] Chapter 10
[Per] Paragraphs 2.1–3.2
[Aru] Chapter 1.1
2. Basics of symplectic geometry. Symplectic linear algebra. Symplectic manifolds. Darboux theorem. Poisson manifolds and symplectic leaves. Symplectic structure on a cotangent bundle of a manifold: Liouville 1-form and canonical symplectic form. Lagrangian submanifolds. Weinstein Lagrangian neighborhood theorem.
[Sil] Chapters 1–2, 18
[BBT] Chapter 14
[AM] Lectures 1–3
[Per] Paragraph 1.3
3. Vector fields on smooth manifolds. One-parameter groups of diffeomorphisms. The graded algebra of differential forms. Derivations of differential forms, contraction with a vector field, and Lie derivative. Magic Cartan's formula. Symplectic and Hamiltonian vector fields on symplectic manifolds.
[Sil] Chapter 18
[BBT] Chapter 14
[AM] Lectures 2–3
4. Lie groups. Smooth actions of Lie groups on smooth manifolds. Left-invariant vector fields, tangent Lie algebra. Maurer–Cartan form. Fundamental vector fields. Adjoint and coadjoint actions of Lie groups. Poisson brackets on a dual to a Lie algebra. Coadjoint orbits of Lie groups, Kirillov–Kostant symplectic structure on coadjoint orbits.
[Kir] Chapters 2–3
[Sil] Chapters 21–22
[AM] Lectures 4–5
5. Symplectic, weakly Hamiltonian, and Hamiltonian actions of Lie groups on symplectic manifolds. Momentum map. Chevalley–Eilenberg complex, Lie algebra cohomologies. Cohomological conditions of existence and uniqueness of the momentum map. Relation to central extensions.
[Sil] Chapters 21–22, 26
[AM] Lectures 5–6
[Wei] Chapter 7
6. Properties of the momentum map. Momentum map as a conservation law for an invariant Hamiltonian. Example: motion in a central field, SO(3) symmetry, angular momentum. Momentum maps for natural actions on cotangent bundles and coadjoint orbits. Classification of coadjoint orbits of unitary group U(n): flag manifolds, Grassmannians, complex projective spaces. Fubini–Study form as a symplectic form on CP^{n - 1}.
[Sil] Chapters 21–22, 26
[AM] Lectures 5–6
7. Actions of compact Lie groups on symplectic manifolds. Regular values of the momentum map. Marsden–Weinstein–Meyer theorem. Symplectic reduction, construction of the symplectic form on the symplectic quotient. Projection method for dynamics given by invariant Hamiltonians. Example: motion in a central field, centrifugal potential.
[Sil] Chapters 23–24
[AM] Lectures 7–8
8. Application of the symplectic reduction method to classical integrable systems. Cotangent bundle of a Lie algebra, reduction with respect to the adjoint Lie group action. Example: unitary group U(n), reduction along minimal nontrivial coadjoint orbit, and the Calogero–Moser system of particles. Lax matrix of the Calogero–Moser system from reduction. Solution of Calogero–Moser equations of motion.
[KKS]
[Per]
9. Cotangent bundle of a Lie group, trivialization by left and right shifts. Iwasawa decomposition. Symplectic reduction with respect to the action of Lie subgroups, integrable Toda chains.
10. Classical r-matrix method. Classical Yang–Baxter equation, modified classical Yang–Baxter equation. Involutivity of Hamiltonians from the classical r-matrix structure. Existence of a classical r-matrix. Classical r-matrices for Calogero–Moser and Toda integrable systems.
11. Lax representation with a spectral parameter. Example: integrable tops. Spin Calogero–Moser system. Classical Yang–Baxter equation with spectral parameters. Permutation matrix and corresponding classical r-matrix. Gaudin integrable model.
12. Integrable systems related to semisimple Lie algebras. Review of Lie algebra theory: root systems and classification of semisimple Lie algebras. Toda chains constructed by a root system. Affine Lie algebras and periodic Toda chain.
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
[Kir] A. Kirillov, Jr. Introduction to Lie groups and Lie algebras. Cambridge University Press, 2010.
[Wei] C. A. Weibel. An introduction to homological algebra. Cambridge University Press, 1995.
[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
[Kir] A. Kirillov, Jr. Introduction to Lie groups and Lie algebras. Cambridge University Press, 2010.
[Wei] C. A. Weibel. An introduction to homological algebra. Cambridge University Press, 1995.
听众
Undergraduate
, Advanced Undergraduate
, Graduate
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笔记公开
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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.