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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 4, Pages 145–156 (Mi izv3989)  

This article is cited in 7 scientific papers (total in 7 papers)

On polynomial integrals of a mechanical system on a two-dimensional torus

A. E. Mironovab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b Novosibirsk State University, Mechanics and Mathematics Department

Abstract: We shall show that if a natural mechanical system defined on a two-dimensional torus and having a real analytic potential possesses a polynomial integral of odd degree in momenta, then the leading coefficients in the momenta satisfy two identities of a special form. We also show that if the system possesses an integral of the fifth degree in momenta, then there exists an integral of the first degree in momenta.

Keywords: integrable Hamiltonian system, polynomial integral.

DOI: https://doi.org/10.4213/im3989

Full text: PDF file (460 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2010, 74:4, 805–817

Bibliographic databases:

UDC: 517.938
MSC: 37J35, 37J99, 37K10, 53C80, 70G10, 70H03, 70H05, 70H06, 70H25
Received: 04.09.2008

Citation: A. E. Mironov, “On polynomial integrals of a mechanical system on a two-dimensional torus”, Izv. RAN. Ser. Mat., 74:4 (2010), 145–156; Izv. Math., 74:4 (2010), 805–817

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Kozlov V.V., “On Gibbs distribution for quantum systems”, p-Adic Numbers Ultrametric Anal. Appl., 4:1 (2012), 76–83  crossref  mathscinet  zmath  scopus
    2. N. V. Denisova, V. V. Kozlov, D. V. Treschev, “Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space”, Izv. Math., 76:5 (2012), 907–921  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. S. V. Agapov, D. N. Alexandrov, “Fourth-Degree Polynomial Integrals of a Natural Mechanical System on a Two-Dimensional Torus”, Math. Notes, 93:5 (2013), 780–783  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. 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
    5. Ivan Yu. Polekhin, “Classical Perturbation Theory and Resonances in Some Rigid Body Systems”, Regul. Chaotic Dyn., 22:2 (2017), 136–147  mathnet  crossref  mathscinet
    6. Thierry Combot, “Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus”, Regul. Chaotic Dyn., 22:4 (2017), 386–497  mathnet  crossref
    7. 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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