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Polynomial integrability of natural two-dimensional Hamiltonian systems and applications

Subject Area Mathematics
Term from 2021 to 2024
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 455806247
 
Final Report Year 2025

Final Report Abstract

The main research direction of our project concerns integability and nonintergability of natural Hamiltonian systems with two degrees of freedom, with the focus on geodesic flows on closed surfaces and on billiards. This is the most classical part of the theory of integrable systems. Within the project, we published or prepublished 23 papers on the topic of the project. Two doctoral students funded by the project submitted their doctoral thesis. Let us highlight the following two results: one is the paper of Bialy et al, published at Annals of Mathematics, with the proof of the Birkhoff-Poritsky conjecture for centrally-symmetric C 2 -smooth convex planar billiards, under mild additional assumptions. Another highlight is the solutions of two conjectures explicitly formulated by Bolsinov, Kozlov and Fomenko, in a recent preprint of Matveev. Three influential conjectures of Bolsinov, Kozlov and Fomenko were explicitly written as the main objectives of the project. We solved two of them, and the solution went along the lines suggested in the application for the project. Let us give more details: the conjectures (b) and (c) explicitly stated by Bolsinov, Kozlov and Fomenko in their joint paper 1995 are closely related and are different versions of the answer to the question whether, on the 2 dimensional sphere, there exists a smooth metric admitting a nontrivial polynomial in momenta integral of an arbitrary large degree k, and admitting no nontrivial polynomial in momenta integral of a lower degree. We have shown that the metrics constructed by K. Kiyohara 2001, which admit irreducible polynomial in momenta integrals of arbitrary high degree, do not admit integrals of lower degree. The main step is the proof is based on real-analyticity of geodesic flows of superintegrable two-dimensional metrics. The real-analyticity was conjectured in the project application, and we indeed proved it under certain additional conditions which Kiyohara’s examples fulfill. As the examples of Kiyohara are not real-analytic, this gives us a contradiction, which shows nonexistence of the additional integral.

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