Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 096, 17 pages      arXiv:2105.11123      https://doi.org/10.3842/SIGMA.2021.096

Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian

Hassan Boualem a and Robert Brouzet b
a) IMAG, Université de Montpellier, France
b) LAMPS, EA 4217, Université Perpignan Via Domitia, France

Received May 24, 2021, in final form October 22, 2021; Published online October 29, 2021

Abstract
We state and prove that a certain class of smooth functions said to be BH-separable is a meagre subset for the Fréchet topology. Because these functions are the only admissible Hamiltonians for Arnold-Liouville systems admitting a bi-Hamiltonian structure, we get that, generically, Arnold-Liouville systems cannot be bi-Hamiltonian. At the end of the paper, we determine, both as a concrete representation of our general result and as an illustrative list, which polynomial Hamiltonians $H$ of the form $H(x,y)=xy+ax^3+bx^2y+cxy^2+dy^3$ are BH-separable.

Key words: completely integrable Hamiltonian system; Arnold-Liouville theorem; action-angle coordinates; bi-Hamiltonian system; separability of functions; change of coordinates; Fréchet topology; meagre set.

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