MAGIC067: Integrable Systems

Course details

A core MAGIC course


Autumn 2022
Monday, October 3rd to Friday, December 9th


Live lecture hours
Recorded lecture hours
Total advised study hours


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. 


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


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. 


  • AV

    Professor Alexander Veselov

    Loughborough University


No bibliography has been specified for this course.


The assessment for this course will be released on Monday 9th January 2023 and is due in by Saturday 21st January 2023 at 23:59.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.


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