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Izv. RAN. Ser. Mat., 2012, Volume 76, Issue 5, Pages 57–72 (Mi izv8001)  

This article is cited in 5 scientific papers (total in 5 papers)

Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space

N. V. Denisovaa, V. V. Kozlovb, D. V. Treschevb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Steklov Mathematical Institute of the Russian Academy of Sciences

Abstract: We consider problems related to the well-known conjecture on the degrees of irreducible polynomial integrals of a reversible Hamiltonian system with two degrees of freedom and toral position space. The main object of study is a special system arising in the analysis of irreducible polynomial integrals of degree 4. In a particular case we have the problem of the motion of two interacting particles on a circle in given potential fields. We prove that if the three potentials are smooth non-constant functions, then this problem has no non-trivial polynomial integrals of arbitrarily high degree. We prove the conjecture completely for systems with a polynomial first integral of degree 4 in the momenta.

Keywords: irreducible integrals, systems with impacts, spectrum of a potential.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00648-а
11-01-12075-офи-м
Ministry of Education and Science of the Russian Federation 11.G34.31.0054
11.G34.31.0039


DOI: https://doi.org/10.4213/im8001

Full text: PDF file (532 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2012, 76:5, 907–921

Bibliographic databases:

Document Type: Article
UDC: 517.9+531.01
MSC: 37J15, 70F05, 70H07
Received: 25.04.2012

Citation: N. V. Denisova, V. V. Kozlov, D. V. Treschev, “Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space”, Izv. RAN. Ser. Mat., 76:5 (2012), 57–72; Izv. Math., 76:5 (2012), 907–921

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. M. Bialy, A. E. Mironov, “Integrable geodesic flows on 2-torus: Formal solutions and variational principle”, J. Geom. Phys., 87 (2015), 39–47  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. I. A. Taimanov, “On first integrals of geodesic flows on a two-torus”, Proc. Steklov Inst. Math., 295 (2016), 225–242  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. Ivan Yu. Polekhin, “Classical Perturbation Theory and Resonances in Some Rigid Body Systems”, Regul. Chaotic Dyn., 22:2 (2017), 136–147  mathnet  crossref  mathscinet
    4. Thierry Combot, “Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus”, Regul. Chaotic Dyn., 22:4 (2017), 386–497  mathnet  crossref
    5. Bolsinov A. Matveev V.S. Miranda E. Tabachnikov S., “Open Problems, Questions and Challenges in Finite-Dimensional Integrable Systems”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2131 (2018), 20170430  crossref  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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