Symmetric Separation of Variables for the Extended Clebsch and Manakov Models
In the present paper, using a modification of the method of vector fields ᵢ of the bi-Hamiltonian theory of separation of variables (SoV), we construct symmetric non-Stäckel variable separation for the three-dimensional extension of the Clebsch model, which is equivalent (in the bi-Hamiltonian sense...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2025 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2025
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/214098 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Symmetric Separation of Variables for the Extended Clebsch and Manakov Models. Taras Skrypnyk. SIGMA 21 (2025), 066, 19 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862588873486368768 |
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| author | Skrypnyk, Taras |
| author_facet | Skrypnyk, Taras |
| citation_txt | Symmetric Separation of Variables for the Extended Clebsch and Manakov Models. Taras Skrypnyk. SIGMA 21 (2025), 066, 19 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In the present paper, using a modification of the method of vector fields ᵢ of the bi-Hamiltonian theory of separation of variables (SoV), we construct symmetric non-Stäckel variable separation for the three-dimensional extension of the Clebsch model, which is equivalent (in the bi-Hamiltonian sense) to the system of interacting Manakov (Schottky-Frahm) and Euler tops. For the obtained symmetric SoV (contrary to the previously constructed asymmetric one), all curves of separation are the same and have genus five. It occurred that the difference between the symmetric and asymmetric cases is encoded in the different form of the vector fields used to construct the separating polynomial. We explicitly construct coordinates and momenta of separation and Abel-type equations in the considered examples of symmetric SoV for the extended Clebsch and Manakov models.
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| first_indexed | 2026-03-21T12:19:55Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-214098 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T12:19:55Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Skrypnyk, Taras 2026-02-19T11:14:40Z 2025 Symmetric Separation of Variables for the Extended Clebsch and Manakov Models. Taras Skrypnyk. SIGMA 21 (2025), 066, 19 pages 1815-0659 2020 Mathematics Subject Classification: 37J35; 37J37; 37J39 arXiv:2508.03107 https://nasplib.isofts.kiev.ua/handle/123456789/214098 https://doi.org/10.3842/SIGMA.2025.066 In the present paper, using a modification of the method of vector fields ᵢ of the bi-Hamiltonian theory of separation of variables (SoV), we construct symmetric non-Stäckel variable separation for the three-dimensional extension of the Clebsch model, which is equivalent (in the bi-Hamiltonian sense) to the system of interacting Manakov (Schottky-Frahm) and Euler tops. For the obtained symmetric SoV (contrary to the previously constructed asymmetric one), all curves of separation are the same and have genus five. It occurred that the difference between the symmetric and asymmetric cases is encoded in the different form of the vector fields used to construct the separating polynomial. We explicitly construct coordinates and momenta of separation and Abel-type equations in the considered examples of symmetric SoV for the extended Clebsch and Manakov models. The author is grateful to Franco Magri for explaining the bi-Hamiltonian approach to separation of variables, for the interest in the work, and for useful discussions. The author is also grateful to the anonymous referees for their corrections, permitting him to improve the text of the article. The research described in this paper was made possible in part by the Isaac Newton Institute and the London Mathematical Society Solidarity grant. The author expresses his gratitude to the grant-givers. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Symmetric Separation of Variables for the Extended Clebsch and Manakov Models Article published earlier |
| spellingShingle | Symmetric Separation of Variables for the Extended Clebsch and Manakov Models Skrypnyk, Taras |
| title | Symmetric Separation of Variables for the Extended Clebsch and Manakov Models |
| title_full | Symmetric Separation of Variables for the Extended Clebsch and Manakov Models |
| title_fullStr | Symmetric Separation of Variables for the Extended Clebsch and Manakov Models |
| title_full_unstemmed | Symmetric Separation of Variables for the Extended Clebsch and Manakov Models |
| title_short | Symmetric Separation of Variables for the Extended Clebsch and Manakov Models |
| title_sort | symmetric separation of variables for the extended clebsch and manakov models |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214098 |
| work_keys_str_mv | AT skrypnyktaras symmetricseparationofvariablesfortheextendedclebschandmanakovmodels |