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.
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. Math., 76:5 (2012), 907–921
\Bibitem{DenKozTre12}
\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. Math.
\yr 2012
\vol 76
\issue 5
\pages 907--921
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Linking options:
https://www.mathnet.ru/eng/im8001
https://doi.org/10.1070/IM2012v076n05ABEH002609
https://www.mathnet.ru/eng/im/v76/i5/p57
This publication is cited in the following 14 articles:
S. V. Agapov, “High-degree polynomial integrals of a natural system on the two-dimensional torus”, Siberian Math. J., 64:2 (2023), 261–268
V. V. Kozlov, “Discrete symmetries of equations of dynamics with polynomial integrals of higher degrees”, Izv. Math., 87:5 (2023), 972–986
S. V. Agapov, M. M. Tursunov, “O ratsionalnykh integralakh dvumernykh naturalnykh sistem”, Sib. matem. zhurn., 64:4 (2023), 665–674
S. V. Agapov, M. M. Tursunov, “On the Rational Integrals of Two-Dimensional Natural Systems”, Sib Math J, 64:4 (2023), 787
Burns K., Matveev V.S., “Open Problems and Questions About Geodesics”, Ergod. Theory Dyn. Syst., 41:3 (2021), PII S0143385719000737, 641–684
N. V. Denisova, “On Momentum-Polynomial Integrals of a Reversible Hamiltonian System of a Certain Form”, Proc. Steklov Inst. Math., 310 (2020), 131–136
S. V. Agapov, “Rational integrals of a natural mechanical system on the 2-torus”, Siberian Math. J., 61:2 (2020), 199–207
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
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
Ivan Yu. Polekhin, “Classical Perturbation Theory and Resonances in Some Rigid Body Systems”, Regul. Chaotic Dyn., 22:2 (2017), 136–147
Thierry Combot, “Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus”, Regul. Chaotic Dyn., 22:4 (2017), 386–497
Leo T. Butler, Lagrangian Mechanics, 2017
I. A. Taimanov, “On first integrals of geodesic flows on a two-torus”, Proc. Steklov Inst. Math., 295 (2016), 225–242
M. Bialy, A. E. Mironov, “Integrable geodesic flows on 2-torus: Formal solutions and variational principle”, J. Geom. Phys., 87 (2015), 39–47
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