Polynomielle Integrabilität von natürlichen zweidimensionalen Hamiltonischen Systemen und deren Anwendungen

• Antragsteller: Professor Dr. Vladimir Matveev
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2021 bis 2024
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 455806247

• Erstellungsjahr: 2025

Zusammenfassung der Projektergebnisse

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.

Projektbezogene Publikationen (Auswahl)

• Dan Reznik’s identities and more. European Journal of Mathematics, 8(4), 1341-1354.
  Bialy, Misha & Tabachnikov, Serge
  
• Billiard Tables with Rotational Symmetry. International Mathematics Research Notices, 2023(5), 3970-4003.
  Bialy, Misha & Tsodikovich, Daniel
  
• Mather β-Function for Ellipses and Rigidity. Entropy, 24(11), 1600.
  Bialy, Michael
  
• Numerical Non-Integrability of Hexagonal String Billiard.
  Bialy, Misha & Youssin, Baruch
  
• Self-Bäcklund curves in centroaffine geometry and Lamé’s equation. Communications of the American Mathematical Society, 2(6), 232-282.
  Bialy, Misha; Bor, Gil & Tabachnikov, Serge
  
• The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables. Annals of Mathematics, 196(1).
  Bialy, Misha & Mironov, Andrey E.
  
• Effective rigidity away from the boundary for centrally symmetric billiards. Ergodic Theory and Dynamical Systems, 44(7), 1741-1756.
  Bialy, Misha
  
• Locally maximising orbits for the non-standard generating function of convex billiards and applications. Nonlinearity, 36(3), 2001-2019.
  Bialy, Misha & Tsodikovich, Daniel
  
• "Integrable geodesic flows with simultaneously diagonalisable quadratic integrals
  Agafonov, Sergei & Matveev, Vladimir Sergeevich
  
• Applications of Nijenhuis Geometry V: Geodesic Equivalence and Finite-Dimensional Reductions of Integrable Quasilinear Systems. Journal of Nonlinear Science, 34(2).
  Bolsinov, Alexey V.; Konyaev, Andrey Yu & Matveev, Vladimir Sergeevich
  
• Integrable outer billiards and rigidity. Journal of Modern Dynamics, 20(0), 51-65.
  Bialy, Misha
  
• Killing tensors on reducible spaces. manuscripta mathematica, 176(1).
  Matveev, Vladimir Sergeevich & Nikolayevsky, Yuri
  
• Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures. Nonlinearity, 37(10), 105003.
  Bolsinov, Alexey V.; Konyaev, Andrey Yu & Matveev, Vladimir Sergeevich
  
• Nijenhuis operators with a unity and F‐manifolds. Journal of the London Mathematical Society, 110(3).
  Antonov, Evgenii I. & Konyaev, Andrey Yu.
  
• On Haantjes tensors for second-order superintegrable systems.
  Marquette, Ian; McLeod, Damien; Scapucci, Serena & Vollmer, Andreas
  
• Orthogonal separation of variables for spaces of constant curvature. Forum Mathematicum.
  Bolsinov, Alexey V.; Konyaev, Andrey Yu & Matveev, Vladimir Sergeevich
  
• Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields. Comptes Rendus. Mathématique, 362(G9), 1043-1049.
  Matveev, Vladimir Sergeevich & Nikolayevsky, Yuri
  
• When a (1,1)-tensor generates separation of variables of a certain metric. Journal of Geometry and Physics, 195, 105031.
  Bolsinov, Alexey V.; Konyaev, Andrey Yu & Matveev, Vladimir Sergeevich
  
• If a Minkowski billiard is projective, then it is the standard billiard. Sbornik: Mathematics, 216(5), 638-653.
  Glutsyuk, Alexey Antonovich & Matveev, Vladimir Sergeevich
  
• Locally maximizing orbits for multi-dimensional twist maps and Birkhoff billiards. Transactions of the American Mathematical Society, 378(6), 4077-4108.
  Bialy, Misha & Tsodikovich, Daniel
  
• On the existence of geodesic vector fields on closed surfaces. Theoretical and Applied Mechanics, 52(1), 109-113.
  Matveev, Vladimir Sergeevich
  
• Real Analyticity of 2-Dimensional Superintegrable Metrics and Solution of Two Bolsinov – Kozlov – Fomenko Conjectures. Regular and Chaotic Dynamics, 30(4), 677-687.
  Matveev, Vladimir Sergeevich
  
• Self-adjoint quantization of Stäckel integrable systems. Journal of Physics A: Mathematical and Theoretical, 58(34), 345202.
  Kress, Jonathan M. & Matveev, Vladimir Sergeevich