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Regular version of the site
2018/2019

Research Seminar "Integrable Systems of Classical Mechanics"

Type: Optional course (faculty)
When: 1, 2 module
Instructors: Ian Marshall
Language: English

Course Syllabus

Abstract

Integrable systems are Hamiltonian systems possessing a complete set of commuting integrals. Since the Kepler problem, which is perhaps the most important physical model in history, and throughout the development of classical mechanics and celestial mechanics, they make recurrent appearances. In modern times they continue to find applications in diverse areas of physics and mathematics, and their study involves many interesting mathematical techniques. In this course we will treat a series of examples by means of which we shall encounter some of the various methods and approaches used in the subject.
Learning Objectives

Learning Objectives

  • The seminar is intended to introduce methods of integrable Hamiltonian systems with emphasis to particular examples.
Expected Learning Outcomes

Expected Learning Outcomes

  • Successful participants will master methods of modern integrable systems theory, be able to apply them to finding explicit solutions of exactly solvable problems of mathematics and mathematical physics.
Course Contents

Course Contents

  • First examples: Jacobi problem of geodesics on an ellipsoid, Neumann problem of harmonic oscillators on a sphere, Euler problem of rigid body motion, Kepler problem, KdV equation.
  • Review of differential geometry: smooth manifold, tangent and cotangent bundles, vector-fields and _p- forms, exterior derivative, Lie derivative, symplectic structure, Poisson structure.
  • Darboux Theorem, generating functions, Liouville Theorem.
  • Euler, problem of two centres: Elliptic coordinates on Lax representation, Garnier and Calogero - Moser systems.
  • The KdV story, and a superficial look at inverse scattering.
  • Lie groups, Lie algebras.
  • Involution theorems, the r-matrix, Toda models, Kowalevski top, Manakov top.
  • Hamiltonian reduction, examples — Calogero and others.
Assessment Elements

Assessment Elements

  • non-blocking Cumulative grade
    cumulative grade is proportional to number of tasks solved.
  • non-blocking Final exam
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    0.3 * Cumulative grade + 0.7 * Final exam
Bibliography

Bibliography

Recommended Core Bibliography

  • Babelon, O., Bernard, D., & Talon, M. (2003). Introduction to Classical Integrable Systems. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=120350
  • Baxter, R. J. (2007). Exactly Solved Models in Statistical Mechanics (Vol. Dover ed). Mineola, N.Y.: Dover Publications. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1152951

Recommended Additional Bibliography

  • Baxter, R. J. (1982). Exactly Solved Models in Statistical Mechanics. [Place of publication not identified]: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1258304