## Announcements

I have uploaded the lecture 4 and 5 and the third problem sheet

Forum

## General

#### Description

This is MAGIC023 module tought by Marta Mazzocco in 2007-2008 who has recently started her work in Loughborough. This course now is offered by Loughborough University. All necessary materials are available on the MAGIC website.
Description This course starts of from the case of finite dimensional Hamiltonian systems. We shall explain what integrability means in this context. We shall introduce the notion of Liouville integrability and state the Arnol'd-Liouville theorem which roughly speaking says that a system is integrable if admits enough (Poisson commuting) constants of motion. We shall then introduce another fundamental concept of modern mathematics: symmetries produce integrals of motion (Emmy Noether's theorem).
Before moving on to infinite dimensional system we shall study the example of the Manakov system trough its Lax pair. We shall show how from the Lax pair it is straightforward to obtain the needed constants of motion to prove integrability. Here too the role of the symmetries in the system is fundamental. This example will lead us to consider the natural integrable systems which live on the coadjoint orbits of a Lie algebra. We shall then adapt this machinery to the case of pseudo-differential operators in order to study infinite dimensional systems such as the KdV equation. If there is enough time we'll study special solutions of the KdV such as solitons, finite gap solution and self-similar solutions.

#### Semester

Spring 2010 (Monday, January 11 to Friday, March 19)

#### Timetable

• Mon 13:05 - 13:55

#### Prerequisites

Analytical mechanics: Hamiltonian and Lagrangian approach. Rigid body equations. Lie algebras: main definitions of Lie algbera and Lie group. Adjoint and co-adjoint action. Differential manifolds: tangent and cotangent bundle, vector fields, differential forms.

#### Syllabus

1) Finite dimensional Hamiltonian systems: Recap on Poisson brackets and canonical transformations. Notion of Liouville integrability. Action angle variables for the pendulum. Arnol'd Liouville theorem (no proof). Noether Theorem Example: solution of the Euler top by elliptic integrals. 2) Hamiltonian systems on coadjoint orbits: Lie algebras. Kostant-Kiriillov Poisson brackets. Lax pairs. Hamiltonian structure of Lax equations. Example: Integrability of the Manakov system on so(n). 3) Infinite dimensional Integrable systems: Pseudo-differential operators. Lax pairs for KdV.

## Lecturer

 Email m.mazzocco@me.com Phone (01509) 22 3187

## Students

 Rikesh Bhatt (Leeds) James Hook (Manchester) Daniel HUGHES (Cardiff) Anton Izosimov (Loughborough) Paul JENNINGS (Leeds) Max Jensen (Durham) Paul Jones (Loughborough) Graham Kemp (Loughborough) Davide Penazzi (Leeds) Veronika Schreiber (Loughborough) Nikola Stoilov (Loughborough) Dragomir Tsonev (Loughborough) Chris Welshman (Manchester) Pumei Zhang (Loughborough)

## Bibliography

 Introduction to classical Integrable systems Babelon, Bernard, Talon

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## Assessment

No assessment information is available yet.

No assignments have been set for this course.

## Files

Files marked L are intended to be displayed on the main screen during lectures.

Week(s) int-syst-lec-1.pdf int-syst-lec-2.pdf int-syst-lec-3.pdf int-syst-lec-4.pdf int-syst-lec-5.pdf int-syst-lec-6.pdf int-syst-lec-7.pdf int-syst-lec-8.pdf int-syst-lec-9.pdf sheet1.pdf sheet2.pdf sheet3.pdf sheet4.pdf sheet5.pdf sheet6.pdf