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I-CELMECH 3-part seminar series

3-part seminar series at the Departement of Mathematics “Tullio Levi-Civita” in Padova, focusing on the KAM theorem and rigorous computer-assisted applications to problems of Celestial Mechanics:

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The challenge of the applicability of KAM theorem: a computer-assisted
approach

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Speaker: Chiara Caracciolo, Uppsala University
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In KAM theory, constructive techniques are essential in order to produce realistic results in physical problems, since there is usually a big gap between purely analytical results and numerical ones. Computer-assisted approaches are an effective way to make these techniques also rigorous. These seminars will be focused on the rigorous use of the normal form method in nearly-integrable Hamiltonian systems.

Seminar 1, Lecture Room 2AB40, 20/2/2024 at 14.00

Proving the KAM theorem using a normal form approach

This seminar will be dedicated to introducing the problem of the
applicability of KAM theorem and the use of normal forms techniques.
Specifically, I will focus on an algorithm that brings a Hamiltonian in
Kolmogorov normal form and that, consequently, can be used to prove the
existence of KAM tori in nearly-integrable Hamiltonian systems.

Seminar 2, Lecture Room 1BC45, 22/2/2024 at 14.30

Computer-assisted proof of the KAM theorem

In this second seminar, I will discuss the use interval arithmetic in order
to make the computations made by a computer rigorous. In particular, I will
stress out what do we need in order to make the proof of existence of KAM tori based on the normal form procedure rigorous and effective. I will describe how to check the hypotheses of the theorem, like the Diophantine condition for the frequency vector and the smallness condition on the perturbing parameter, using a set of computer-assisted estimates on the Hamiltonian function.

Seminar 3, Lecture Room 2AB40, 27/2/2024 at 14.00

Some applications of a computer-assisted KAM theorem to planetary
three-body problems

This last seminar is devoted to showing some example of rigorous application of the KAM theorem to realistic planetary problems. The first results concern a secular three-body problem, with parameters that are compatible with the observations of the Upsilon-Andromedae planetary system. This system is rather peculiar in view of its non-coplanar nature (with mutual inclination of around 30 degrees) and relatively large eccentricities (approximately 0.3). Secondly, I will present ongoing work in the framework of a realistic planar three-body problem. In particular, I will emphasize what is necessary in order to obtain a positive outcome, by providing examples of both successes and failures, depending on different parameters.

The results presented in these seminars are based on collaborations with U. Locatelli, M. Sansottera and M. Volpi, supported by the PRIN project “New Frontiers of Celestial Mechanics: theory and Applications”.