THE DIFFIETY INSTITUTE SCHOOLS
1-st order PDE's with one unknown function
Valeriy A. YUMAGUZHIN
A program of the course at the Join Russian-Italian Diffiety School,
Pereslavl-Zalessky (Russia), August 17  -  30, 1999
  1. The jet-bundle J1M
    The Cartan distribution on J1M. The canonical form U1. The contact structure on J1M. Integral manifolds of the Cartan distribution.
  2. Contact transformations
    Contact transformations.
    Contact vector fields and their flows. Generating functions.
    The Jacobi bracket.
  3. Classical symmetries
    The Cartan distribution on an equation. Infinitesimal symmetries.
    Trivial symmetries.
  4. The Cauchy problem
    Characteristics and solutions. The Cauchy problem.
  5. Hamilton-Jacobi equations
    The symplectic structure on T*M. Hamiltonian vector fields. The Poisson bracket. The Cauchy problem for Hamilton-Jacobi equations.
  6. Integrable Hamiltonian equations
    1-st integrals.
    The Liouville theorem for integrable systems.



References

[]
I.S. Krasil'shchik and A.M. Vinogradov, Eds, Symmetries and conservation laws for differential equations of mathematical physics, Translations of Mathematical Monographs, Vol. 182, 1999, American Mathematical Society.
[]
V.I. Arnold, Additional chapters of the theory of ordinary differential equations, Moscow, "Nauka", 1978, 304 pp. (Russian).
[]
A.M. Vinogradov and B.A. Kupershmidt, The structure of Hamiltonian mechanics, Russian Math. Surveys, 32, No 4, (1977), 175-236. (Russian).
[]
V.I. Arnold, Mathematical methods of classical mechanics, Moscow, "Nauka", 1989, 472 p. (Russian)
[]
V.V. Kozlov, Symmetries, topology, and resonances in the Hamiltonian mechanics, Publishing house of the Izhevsk state university, Izhevsk, 1995, 432 p. (Russian)