Joseph Krasil'shchik's introductory lectures on the Geometry of Differential Equations

Autumn 2010 - Spring 2011

Lectures take place at the Independent University of Moscow on Wednesday evenings from 17:30 to 19:10 in room 206

Short syllabus

• Jets of bundles
  The Cartan distribution
  Symmetries of the Cartan distribution
  The Lie-Bäcklund theorem

• Nonlinear differential equations and operators
  Classical symmetries of differential equations, their computation and application
  The Bianchi-Lie theorem

• Infinite jets and geometric structures on them
  The Cartan distribution on infinite jets and its symmetries
  Linearizations and evolutionary vector fields

• Prolongations of differential equations
  Infinitely prolonged equations
  Higher symmetries of differential equations, their computation and application

• Recursion operators for symmetries
  Nijenhuis brackets

• Differential coverings and nonlocal symmetries

• Lagrangian and Hamiltonian formalism on infinite jets
  Schouten brackets
  Variational complex

• Lagrangian and Hamiltonian formalism with constraints
  The Vinogradov spectral sequence

Video records of the lectures

Via http://erb-files.narod.ru

• Lecture 1 (15 September 2010)

• Lecture 2 (22 September 2010)

• Lecture 3 (6 October 2010)

• Lecture 4 (20 October 2010)

• Lecture 5 (27 October 2010)

• Lecture 6 (3 November 2010)

• Lecture 7 (10 November 2010)

• Lecture 8 (17 November 2010)

Recommended literature

A. V. Bocharov, A. M. Verbovetsky, A. M. Vinogradov (editor), S. V. Duzhin, I. S. Krasil'shchik (editor), A. V. Samokhin, Yu. N. Torkhov, N. G. Khor'kova, and V. N. Chetverikov Simmetrii i zakony sokhraneniya uravnenij matematicheskoj fiziki, Factorial, Moscow, 2nd edition, 2005.

(English translation: Symmetries and conservation laws for differential equations of mathematical physics, AMS, 1999.)