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arXiv · 2105.11123

Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian

Hassan Boualem·Robert Brouzet
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Abstract

We state and prove that a certain class of smooth functions said to be BH-separable is a meagre subset for the Fr\'echet 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.

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Hassan Boualem, Robert Brouzet. 2021-05-24. Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian. https://doi.org/10.3842/sigma.2021.096

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