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Institut für Mathematik

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# Mathematical Physics 3:

## Integrable Systems of Classical Mechanics (WS 2016/2017)

 Lectures Matteo Petrera 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

• Lecture 1. Smooth manifolds (defs., examples, tangent and cotangent bundles, 1-forms, vector fields)

• 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)

• Lecture 3. Distributions (defs., integral manifolds, integrability of a distribution, Frobenius Thm, foliations), k-forms (defs., exterior algebra, De Rham complex, interior product)

• 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)

• Lecture 5. Lie groups and Lie algebras (defs. and facts, subgroups, subalgebras), matrix Lie groups and matrix Lie algebras (defs., Ado Thm, examples)

• 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)

• Lecture 7. Adjoint actions (invariant functions), quadratic Lie algebras (defs. and facts, Killing form)

• 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)

• 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

• Lecture 10. Rank of a Poisson manifold, Poisson involutive and independent functions, examples (constant Poisson structure, canonical Poisson structure)

• Lecture 11. Symplectic manifolds (construction of a Poisson structure, symplectic morphisms, Darboux Thm, examples), foliation of Poisson manifolds

• 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

• 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)

• Lecture 14. Tensor formulation of the Lie-Poisson bracket, complete integrability of Hamiltonian systems

• Lecture 15. Bi-Hamiltonian systems, solvability by quadratures (general construction, examples)

• Lecture 16. Arnold-Liouville theorem

• 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)

• Lecture 18. R-matrix formalism (Semenov-Tian-Shansky Thm, Adler-Kostant-Symes Thm, linear R-bracket on a Lie algebra)

• Lecture 19. R-matrix formalism (Lax hierarchies and commuting flows), r-matrix formalism (Lie bialgebras, coboundary Lie bialgebras, r-matrix)

• 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)

• Lecture 21. r-matrix formalism (Babelon-Viallet Thm, tensor formulation of the linear R-bracket), Examples (open-end Toda lattice)

### Policy

• This course is not supported by a Tutorial.
• An oral exam will be offered at the end of the semester in case of presence at minimum 75% of the course.

 Matteo Petrera . 18.01.2017.