Abstract:
Under study are the problems of existence of the first integrals rational in momenta for a natural mechanical system on the 2-torus. We prove that a rational integral with a linear numerator and denominator reduces to a linear integral. Considering the case of a quadratic numerator and linear denominator, we obtain an analogous result under some additional assumptions on the coefficients of the first integral.
Keywords:
natural mechanical system, rational first integral.
Citation:
S. V. Agapov, “Rational integrals of a natural mechanical system on the 2-torus”, Sibirsk. Mat. Zh., 61:2 (2020), 255–265; Siberian Math. J., 61:2 (2020), 199–207
\Bibitem{Aga20}
\by S.~V.~Agapov
\paper Rational integrals of a~ natural mechanical system on the 2-torus
\jour Sibirsk. Mat. Zh.
\yr 2020
\vol 61
\issue 2
\pages 255--265
\mathnet{http://mi.mathnet.ru/smj5979}
\crossref{https://doi.org/10.33048/smzh.2020.61.202}
\elib{https://elibrary.ru/item.asp?id=43305768}
\transl
\jour Siberian Math. J.
\yr 2020
\vol 61
\issue 2
\pages 199--207
\crossref{https://doi.org/10.1134/S0037446620020020}
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Linking options:
https://www.mathnet.ru/eng/smj5979
https://www.mathnet.ru/eng/smj/v61/i2/p255
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Sergey I. Agafonov, Thaís G. P. Alves, “Fractional-linear integrals of geodesic flows on surfaces and Nakai's geodesic 4-webs”, Advances in Geometry, 24:2 (2024), 263
S. V. Agapov, M. M. Tursunov, “O ratsionalnykh integralakh dvumernykh naturalnykh sistem”, Sib. matem. zhurn., 64:4 (2023), 665–674
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S. V. Agapov, M. M. Tursunov, “On the Rational Integrals of Two-Dimensional Natural Systems”, Sib Math J, 64:4 (2023), 787
S. Agapov, V. Shubin, “Rational integrals of 2-dimensional geodesic flows: new examples”, J. Geom. Phys., 170 (2021), 104389
S. V. Agapov, “On first integrals of two-dimensional geodesic flows”, Siberian Math. J., 61:4 (2020), 563–574
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