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Mat. Sb., 2000, Volume 191, Number 2, Pages 43–63 (Mi msb452)  

This article is cited in 17 scientific papers (total in 18 papers)

Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space

N. V. Denisova, V. V. Kozlov

M. V. Lomonosov Moscow State University

Abstract: The problem considered here is that of finding conditions ensuring that a reversible Hamiltonian system has integrals polynomial in momenta. The kinetic energy is a zero-curvature Riemannian metric and the potential a smooth function on a two-dimensional torus. It is known that the existence of integrals of degrees 1 and 2 is related to the existence of cyclic coordinates and the separation of variables. The following conjecture is also well known: if there exists an integral of degree $n$ independent of the energy integral, then there exists an additional integral of degree 1 or 2. In the present paper this result is established for $n=3$ (which generalizes a theorem of Byalyi), and for $n=4$, $5$, and $6$ this is proved under some additional assumptions about the spectrum of the potential.


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English version:
Sbornik: Mathematics, 2000, 191:2, 189–208

Bibliographic databases:

UDC: 517.9+531.01
MSC: 58F05, 70H05
Received: 21.06.1999

Citation: N. V. Denisova, V. V. Kozlov, “Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space”, Mat. Sb., 191:2 (2000), 43–63; Sb. Math., 191:2 (2000), 189–208

Citation in format AMSBIB
\by N.~V.~Denisova, V.~V.~Kozlov
\paper Polynomial integrals of reversible mechanical systems with a~two-dimensional torus as the~configuration space
\jour Mat. Sb.
\yr 2000
\vol 191
\issue 2
\pages 43--63
\jour Sb. Math.
\yr 2000
\vol 191
\issue 2
\pages 189--208

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    This publication is cited in the following articles:
    1. V. V. Kozlov, “Sofya Kovalevskaya: a mathematician and a person”, Russian Math. Surveys, 55:6 (2000), 1175–1192  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. Dullin H.R., Matveev V.S., “A new integrable system on the sphere”, Math. Res. Lett., 11:5-6 (2004), 715–722  crossref  mathscinet  zmath  isi  elib
    3. Kozlov V.V., Treshchev D.V., “Conservation laws in quantum systems on a torus”, Dokl. Math., 70:2 (2004), 807–810  mathnet  mathscinet  isi  isi  elib
    4. Dullin H.R., Matveev V.S., “A new natural Hamiltonian system on T*S-2 admitting an integral of degree 3 in momenta”, Global Analysis and Applied Mathematics, Aip Conference Proceedings, 729, 2004, 141–146  crossref  mathscinet  zmath  adsnasa  isi
    5. Rudnev M., Ten V., “A model for separatrix splitting near multiple resonances”, Regul. Chaotic Dyn., 11:1 (2006), 83–102  mathnet  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    6. Rylov A.I., “Infinite set of polynomial conservation laws in gas dynamics”, Dokl. Math., 76:3 (2007), 962–964  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    7. Kozlov V.V., “Several problems on dynamical systems and mechanics”, Nonlinearity, 21:9 (2008), T149–T155  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    8. A. E. Mironov, “On polynomial integrals of a mechanical system on a two-dimensional torus”, Izv. Math., 74:4 (2010), 805–817  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. V. V. Kozlov, “On Gibbs distribution for quantum systems”, P-Adic Num Ultrametr Anal Appl, 4:1 (2012), 76  crossref  mathscinet  zmath  scopus  scopus  scopus
    10. 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
    11. 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
    12. V. S. Kalnitsky, “Symmetries of a flat cosymbol algebra of the differential operators”, J. Math. Sci. (N. Y.), 222:4 (2017), 429–436  mathnet  crossref  mathscinet
    13. 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
    14. Ivan Yu. Polekhin, “Classical Perturbation Theory and Resonances in Some Rigid Body Systems”, Regul. Chaotic Dyn., 22:2 (2017), 136–147  mathnet  crossref  mathscinet
    15. Thierry Combot, “Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus”, Regul. Chaotic Dyn., 22:4 (2017), 386–497  mathnet  crossref
    16. 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  mathscinet  zmath  isi
    17. 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
    18. 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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