Rational Integrals of a Natural Mechanical System on the 2-Torus

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.

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Acknowledgments

The author thanks V. V. Shubin for helpful discussions.

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Correspondence to S. V. Agapov.

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Russian Text © The Author(s), 2020, published in Sibirskii Matematicheskii Zhurnal, 2020, Vol. 61, No. 2, pp. 255–265.

The author was supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-5913.2018.1).

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Agapov, S.V. Rational Integrals of a Natural Mechanical System on the 2-Torus. Sib Math J 61, 199–207 (2020). https://doi.org/10.1134/S0037446620020020

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Keywords

  • natural mechanical system
  • rational first integral