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
- The jet-bundle J^{1}M
The Cartan distribution on J^{1}M. The canonical form U_{1}.
The contact structure on J^{1}M. Integral manifolds of the Cartan
distribution.
- Contact transformations
Contact transformations.
Contact vector fields and their flows. Generating functions.
The Jacobi bracket.
- Classical symmetries
The Cartan distribution on an equation. Infinitesimal symmetries.
Trivial symmetries.
- The Cauchy problem
Characteristics and solutions. The Cauchy problem.
- Hamilton-Jacobi equations
The symplectic structure on T^{*}M. Hamiltonian vector fields.
The Poisson bracket. The Cauchy problem for Hamilton-Jacobi equations.
- 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)
Questions and suggestions should go to
Jet Nestruev, jet @ diffiety.ac.ru.