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Generically, Arnold–Liouville Systems Cannot be Bi-Hamiltonian
Hassan Boualema, Robert Brouzetb a IMAG, Université de Montpellier, France
b LAMPS, EA 4217, Université Perpignan Via Domitia, France
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.
Keywords:
completely integrable Hamiltonian system, Arnold–Liouville theorem, action-angle coordinates, bi-Hamiltonian system, separability of functions, change of coordinates, Fréchet topology, meagre set.
Received: May 24, 2021; in final form October 22, 2021; Published online October 29, 2021
Citation:
Hassan Boualem, Robert Brouzet, “Generically, Arnold–Liouville Systems Cannot be Bi-Hamiltonian”, SIGMA, 17 (2021), 096, 17 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1778 https://www.mathnet.ru/eng/sigma/v17/p96
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Abstract page: | 51 | Full-text PDF : | 12 | References: | 12 |
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