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This course is part of the MAGIC core.


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.


Autumn 2018 (Monday, October 8 to Friday, December 14)


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


  • Wed 11:05 - 11:55
  • Wed 12:05 - 12:55


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.


Alexander Veselov
Phone (01509) 222866
Photo of Alexander Veselov


Mathematical Methods of Classical MechanicsArnol'd
Integrable Systems of Classical Mechanics and Lie AlgebrasPerelomov
Introduction to Classical Integrable SystemsBabelon, Bernard and Talon


Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)


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.

Exam MAGIC067 2018

Files:Exam paper
Released: Sunday 6 January 2019 (42.9 days ago)
Deadline: Sunday 20 January 2019 (27.9 days ago)

Please, answer all the questions, you will need the equivalent of 2 questions to pass. Write your answers/solutions, scan them and upload to the MAGIC web page. Good luck!

Recorded Lectures

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