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

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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.

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V.I. Arnold, Additional chapters of the theory of ordinary differential equations, Moscow, "Nauka", 1978, 304 pp. (Russian).

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A.M. Vinogradov and B.A. Kupershmidt, The structure of Hamiltonian mechanics, Russian Math. Surveys, 32, No 4, (1977), 175-236. (Russian).

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V.I. Arnold, Mathematical methods of classical mechanics, Moscow, "Nauka", 1989, 472 p. (Russian)

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V.V. Kozlov, Symmetries, topology, and resonances in the Hamiltonian mechanics, Publishing house of the Izhevsk state university, Izhevsk, 1995, 432 p. (Russian)