Geodesic
Generically, Arnold–Liouville Systems Cannot be Bi-Hamiltonian
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: completely integrable Hamiltonian system, Arnold–Liouville theorem, action-angle coordinates, bi-Hamiltonian system, separability of functions, change of coordinates, meagre set.
Mots-clés : Fréchet topology 
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     title = {Generically, {Arnold{\textendash}Liouville} {Systems} {Cannot} be {Bi-Hamiltonian}},
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Hassan Boualem; Robert Brouzet. Generically, Arnold–Liouville Systems Cannot be Bi-Hamiltonian. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a95/

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