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

This article is cited in 8 scientific papers (total in 8 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
Ministry of Education and Science of the Russian Federation 11.G34.31.0054

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

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

Bibliographic databases:

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
\by N.~V.~Denisova, V.~V.~Kozlov, D.~V.~Treschev
\paper Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space
\jour Izv. RAN. Ser. Mat.
\yr 2012
\vol 76
\issue 5
\pages 57--72
\jour Izv. Math.
\yr 2012
\vol 76
\issue 5
\pages 907--921

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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. 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
    6. Agapov S., Valyuzhenich A., “Polynomial Integrals of Magnetic Geodesic Flows on the 2-Torus on Several Energy Levels”, Discret. Contin. Dyn. Syst., 39:11 (2019), 6565–6583  crossref  isi
    7. N. V. Denisova, “On Momentum-Polynomial Integrals of a Reversible Hamiltonian System of a Certain Form”, Proc. Steklov Inst. Math., 310 (2020), 131–136  mathnet  crossref  crossref  mathscinet  isi  elib
    8. S. V. Agapov, “Ratsionalnye integraly naturalnoi mekhanicheskoi sistemy na dvumernom tore”, Sib. matem. zhurn., 61:2 (2020), 255–265  mathnet  crossref
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