# Rational integrability of trigonometric polynomial potentials on the flat torus

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.
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https://hal.archives-ouvertes.fr/hal-01591576
Contributor : Imb - Université de Bourgogne <>
Submitted on : Thursday, September 21, 2017 - 3:29:23 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM

### Citation

Thierry Combot. Rational integrability of trigonometric polynomial potentials on the flat torus. Regular and Chaotic Dynamics, MAIK Nauka/Interperiodica, 2017, 22 (4), pp.386 - 407. ⟨10.1134/S1560354717040049⟩. ⟨hal-01591576⟩

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