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General


This 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.

Semester

Autumn 2016 (Monday, October 3 to Friday, December 9)

Timetable

  • Tue 14:05 - 14:55
  • Tue 15:05 - 15:55

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


Alexander Veselov
Email A.P.Veselov@lboro.ac.uk
Phone (01509) 222866
vcard
Photo of Alexander Veselov


Students


Photo of Kholood Alnefaie
Kholood Alnefaie
(Sheffield)
Photo of Jinrong Bao
Jinrong Bao
(Loughborough)
Photo of John Blackman
John Blackman
(Durham)
Photo of Maxime Fairon
Maxime Fairon
(Leeds)
Photo of Allan Gerrard
Allan Gerrard
(York)
Photo of Thomas Honey
Thomas Honey
(Manchester)
Photo of Amal Mohammed
Amal Mohammed
(*Ext_Assessed)
Photo of Junaid Mustafa
Junaid Mustafa
(*Ext_Assessed)
Photo of Pedro Peres
Pedro Peres
(Exeter)
Photo of Gregory Roberts
Gregory Roberts
(Liverpool)
Photo of Joakim Stromvall
Joakim Stromvall
(Surrey)
Photo of Yiru Ye
Yiru Ye
(Loughborough)
Photo of Guangxiu Zhao
Guangxiu Zhao
(Loughborough)


Bibliography


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


Note:

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Assessment



The 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

Files:Exam paper
Released: Monday 9 January 2017 (44.6 days ago)
Deadline: Sunday 22 January 2017 (30.6 days ago)
Instructions:

Answer all 4 questions, presenting the calculations in detail. The pass mark is 50



Files


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Recorded Lectures


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