"The KAM theorem, following the original
approach designed by Kolmogorov"
A minicourse on KAM theory, delivered in the framework of the Pisa-Hokkaido-Roma2 school on Mathematics and its applications Contents of the lectures
A short introduction to the Hamiltonian formalism.
Poisson brackets. Canonical transformations.
Ref.: Chap. 1
and Chap. 2,
making part of the notes
about "Hamiltonian
Systems", written
by A. Giorgilli.
A short discussion about the Liouville theorem and its
extension provided by Arnold and Jost.
Ref.: Sections 3.2 and 3.5 of Chap. 3,
making part of the notes
about "Hamiltonian
Systems", written
by A. Giorgilli.
Introduction of the general problem of the dynamics.
Poincaré theorem on the non-existence of first integrals
(sketch of the proof, discussion of the consequences).
Ref.: Sections 4.3 and 4.4 of Chap. 4,
making part of the notes
about "Hamiltonian
Systems", written
by A. Giorgilli.
Statement of the KAM theorem. Main ideas behind it.
Kolmogorov normal form.
Ref.: Sections 7.1 and 7.2.2 of Chap. 7,
making part of the notes
about "Hamiltonian
Systems", written
by A. Giorgilli.
Lie series as a useful tool for defining near to
identity canonical transformations: a purely formal
introduction. Discussion of the exchange theorem (by
Gröbner).
Ref.: Sections 6.1, 6.2.1 and 6.2.2 of Chap. 6,
making part of the notes
about "Hamiltonian
Systems", written
by A. Giorgilli.
Analytical settings that are necessary to study the
convergence of Lie series: complexified domains, weighted
Fourier norms, estimates for (multiple) Poisson
brackets.
Ref.: Sections 6.4.2 and 6.7 of Chap. 6,
making part of the notes
about "Hamiltonian
Systems", written
by A. Giorgilli.
Formal algorithm constructing the Kolmogorov normal
form, by using the convergence method typical of the
classical series. Accumulation of the small divisors (short
discussion).
Ref.: Section 7.2.3 of Chap. 7,
making part of the notes
about "Hamiltonian
Systems", written
by A. Giorgilli.
Proof of the KAM theorem, by using the so called
quadratic (or Newton-like or super-convergent) method:
reformulation of the algorithm constructing the Kolmogorov
normal form, iterative lemma, estimates about the smallness
of the perturbation.
Ref.: Section 7.5 of Chap. 7,
making part of the notes
about "Hamiltonian
Systems", written
by A. Giorgilli.
Relaxing the hypotheses of the KAM theorem.
A short and partial overview of the state-of-the-art
in the scientific literature.