MAGIC067: Integrable Systems

Course details

A core MAGIC course

Semester

Autumn 2018
Monday, October 8th to Friday, December 14th

Hours

Live lecture hours
20
Recorded lecture hours
0
Total advised study hours
80

Timetable

Wednesdays
11:05 - 11:55
Wednesdays
12:05 - 12:55

Description

The course is an introduction to the theory of integrable systems. We will consider mainly the finite-dimensional Hamiltonian systems with integrability understood in Liouville's sense. The content covers both classical techniques like separation of variables in the Hamilton-Jacobi equation as well as modern inverse spectral transform method. The main examples include Kepler problem, geodesic flow on ellipsoids, Euler top, Toda lattice, Calogero-Moser system and Korteweg- de Vries equation.

Prerequisites

Students are advised to attend the MAGIC courses on Differentiable Manifolds 063 and on Lie Groups and Lie Algebras 008.

Syllabus

Hamiltonian systems and Poisson brackets. Integrals and symmetries, Noether principle. Example: Kepler system.
Integrability in Liouville’s sense. Liouville-Arnold theorem, action-angle variables. Example: anisotropic harmonic oscillator.
Hamilton-Jacobi equation and separation of variables. Geodesics on ellipsoids and Jacobi inversion problem for hyperelliptic integrals.
Euler equations on Lie algebras and coadjoint orbits. Multidimensional Euler top, Manakov’s generalisation and Lax representation.
Toda lattice and inverse spectral transform method. Direct and inverse spectral problems for Jacobi matrices and explicit solution to open Toda lattice.
Calogero-Moser system and Hamiltonian reduction. Scattering in Calogero-Moser system.
Korteweg-de Vries equation as an infinite-dimensional integrable system. Integrals and Hamiltonian structures, Lenard-Magri scheme.

Lecturer

  • AV

    Professor Alexander Veselov

    University
    Loughborough University

Bibliography

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Assessment

Description

The assessment for this course will be via a single take-home paper in January with 2 weeks to complete and submit online. There will be 4 questions and you will need the equivalent of 2 questions to pass.

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Files

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Lectures

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