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Regul. Chaotic Dyn., 2017, Volume 22, Issue 4, Pages 386–497 (Mi rcd262)  

This article is cited in 2 scientific papers (total in 2 papers)

Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus

Thierry Combot

Scuola Normale Superiore, Centro di Ricerca Matematica Ennio De Giorgi, Laboratorio Fibonacci, Piazza Cavalieri, 56127 Pisa

Abstract: We consider a lattice $\mathcal{L}\subset \mathbb{R}^n$ and a trigonometric potential $V$ with frequencies $k\in\mathcal{L}$. We then prove a strong rational integrability condition on $V$, using the support of its Fourier transform. We then use this condition to prove that a real trigonometric polynomial potential is rationally integrable if and only if it separates up to rotation of the coordinates. Removing the real condition, we also make a classification of rationally integrable potentials in dimensions $2$ and $3$ and recover several integrable cases. After a complex change of variables, these potentials become real and correspond to generalized Toda integrable potentials. Moreover, along the proof, some of them with high-degree first integrals are explicitly integrated.

Keywords: trigonometric polynomials, differential Galois theory, integrability, Toda lattice

DOI: https://doi.org/10.1134/S1560354717040049

References: PDF file   HTML file

Bibliographic databases:

MSC: 37J30
Received: 27.04.2017
Accepted:01.06.2017
Language:

Citation: Thierry Combot, “Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus”, Regul. Chaotic Dyn., 22:4 (2017), 386–497

Citation in format AMSBIB
\Bibitem{Com17}
\by Thierry Combot
\paper Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus
\jour Regul. Chaotic Dyn.
\yr 2017
\vol 22
\issue 4
\pages 386--497
\mathnet{http://mi.mathnet.ru/rcd262}
\crossref{https://doi.org/10.1134/S1560354717040049}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85026864525}


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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. S. V. Agapov, “Ratsionalnye integraly naturalnoi mekhanicheskoi sistemy na dvumernom tore”, Sib. matem. zhurn., 61:2 (2020), 255–265  mathnet  crossref
    2. S. V. Agapov, “O pervykh integralakh dvumernykh geodezicheskikh potokov”, Sib. matem. zhurn., 61:4 (2020), 721–734  mathnet  crossref
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