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Mathematical Physics 3:
Integrable Systems of Classical Mechanics (WS 2016/2017)
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Lectures |
Matteo Petrera
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Tue |
12:15 - 13:45 |
MA 551 |
Thu |
12:15 - 13:45 |
MA 848 |
Office hours |
Mon |
12:00 - 13:00 |
MA 819 |
This is a course of the
Berlin Mathematical School
held in English.
Contents
Manifolds, Lie groups, Lie algebras, Poisson structures, Lie-Poisson structures,
Completely integrable systems, R-brackets, r-brackets, Examples
News
First 2017 Lecture on 05.01
Lectures
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Lecture 1. Smooth manifolds
(defs., examples, tangent and cotangent bundles, 1-forms, vector fields)
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Lecture 2. Vector fields (defs., integral curves, flows, infinitesimal generators, Lie algebra of vector fields,
Picard-Lindelöf Thm, Lie Thm),
maps between manifolds (defs., differential and rank of a map, classical Thms, submanifolds, examples)
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Lecture 3. Distributions (defs., integral manifolds, integrability of a distribution,
Frobenius Thm, foliations), k-forms
(defs., exterior algebra, De Rham complex, interior product)
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Lecture 4. k-forms (pull-back, volume/measure preserving maps), k-vector fields (defs., examples), Lie derivatives
(defs., Cartan formula, commuting vector fields, invariant functions)
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Lecture 5. Lie groups and Lie algebras
(defs. and facts, subgroups, subalgebras), matrix Lie groups and matrix Lie algebras (defs., Ado Thm, examples)
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Lecture 6. Lie group actions
(defs. and facts, generalization to exterior algebra, examples, orbits, facts), representations (defs., coboundary operator),
adjoint actions (defs., generalization to tensor algebra)
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Lecture 7. Adjoint actions (invariant functions), quadratic Lie algebras
(defs. and facts, Killing form)
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Lecture 8.
Quadratic Lie algebras
(Killing form, simple and semi-simple Lie algebras, examples),
loop algebras, Poisson manifolds (Poisson brackets,
Hamiltonian vector fields, Casimirs, Poisson 2-vector fields, Schouten-Nijenhuis bracket)
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Lecture 9. Poisson vector
fields, Poisson morphisms, Hamiltonian actions, Noether Thm, multi-Hamiltonian manifolds and vector fields, local
representation of a Poisson structure, local Hamilton equations
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Lecture 10. Rank of a Poisson manifold, Poisson involutive
and independent functions, examples (constant Poisson structure, canonical Poisson
structure)
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Lecture 11. Symplectic manifolds (construction of a Poisson structure,
symplectic morphisms, Darboux Thm, examples), foliation of Poisson manifolds
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Lecture 12. Foliation of Poisson manifolds (Weinstein Thm, geometric
properties of the symplectic leaves, examples), Lie-Poisson structure on the dual of a Lie algebra
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Lecture 13. Lie-Poisson structures on the dual of a Lie algebra and on the Lie
algebra (defs., Hamiltonian vector fields, Kostant-Kirillov Thm, examples,
Casimirs, spectral invariants, Lax equations)
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Lecture 14. Tensor formulation of the Lie-Poisson bracket, complete integrability of Hamiltonian
systems
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Lecture 15. Bi-Hamiltonian systems, solvability by quadratures (general
construction, examples)
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Lecture 16. Arnold-Liouville theorem
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Lecture 17. Existence of action-angle variables,
R-matrix formalism (Lax equations, linear R-bracket on the dual of a Lie algebra,
R-matrix, double Lie algebras, Yang-Baxter equation, Lie algebra splitting and R-matrices)
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Lecture 18. R-matrix formalism
(Semenov-Tian-Shansky Thm, Adler-Kostant-Symes Thm, linear R-bracket on a Lie algebra)
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Lecture 19.
R-matrix formalism (Lax hierarchies and commuting flows),
r-matrix formalism (Lie bialgebras, coboundary Lie bialgebras, r-matrix)
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Lecture 20.
r-matrix formalism (conditions to define coboundary Lie bialgebras, algebraic Schouten bracket,
Yang-Baxter equation, linear r-bracket on a Lie algebra,
tensor formulation, relation between R-matrices and r-matrices, Lax operators)
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Lecture 21.
r-matrix formalism (Babelon-Viallet Thm, tensor formulation of the linear R-bracket),
Examples (open-end Toda lattice)
Policy
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This course is not supported by a Tutorial.
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An oral exam will be offered at the end of the semester
in case of presence at minimum 75% of the course.
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