Integrable Systems (MAGIC067) |
GeneralThis course is part of the MAGIC core. 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.
SemesterAutumn 2016 (Monday, October 3 to Friday, December 9) Timetable
PrerequisitesStudents are advised to attend the MAGIC courses on Differentiable Manifolds 063 and on Lie Groups and Lie Algebras 008.
SyllabusHamiltonian 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.
Students
Bibliography
Note: 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.) AssessmentThe assessment will be via a take-home exam (4 questions), which must be completed during the assessment period 9-22 January 2017.
The pass mark is 50
Exam paper
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