Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian
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-Liouvill...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2021
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211431 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian. Hassan Boualem and Robert Brouzet. SIGMA 17 (2021), 096, 17 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862729105951162368 |
|---|---|
| author | Boualem, Hassan Brouzet, Robert |
| author_facet | Boualem, Hassan Brouzet, Robert |
| citation_txt | Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian. Hassan Boualem and Robert Brouzet. SIGMA 17 (2021), 096, 17 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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 of the form (, ) = + ³+²+²+³ are BH-separable.
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| first_indexed | 2026-04-17T14:41:46Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211431 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T14:41:46Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Boualem, Hassan Brouzet, Robert 2026-01-02T08:31:43Z 2021 Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian. Hassan Boualem and Robert Brouzet. SIGMA 17 (2021), 096, 17 pages 1815-0659 2020 Mathematics Subject Classification: 26A21; 26B35; 26B40; 37J35; 37J39; 58K15; 70H06 arXiv:2105.11123 https://nasplib.isofts.kiev.ua/handle/123456789/211431 https://doi.org/10.3842/SIGMA.2021.096 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 of the form (, ) = + ³+²+²+³ are BH-separable. We thank Timothy Neal for his proofreading and for improving the English language of our paper. We also thank Roman G. Smirnov for drawing our attention to the importance of Lenard’s early work in the genesis of the theory of bi-Hamiltonian systems. Finally, we thank the anonymous referees for their insightful comments and careful reading, which greatly improved this article. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian Article published earlier |
| spellingShingle | Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian Boualem, Hassan Brouzet, Robert |
| title | Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian |
| title_full | Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian |
| title_fullStr | Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian |
| title_full_unstemmed | Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian |
| title_short | Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian |
| title_sort | generically, arnold-liouville systems cannot be bi-hamiltonian |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211431 |
| work_keys_str_mv | AT boualemhassan genericallyarnoldliouvillesystemscannotbebihamiltonian AT brouzetrobert genericallyarnoldliouvillesystemscannotbebihamiltonian |