On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems
Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show tha...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2015 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2015
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/147160 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems / M. Santoprete // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 15 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862673100542312448 |
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| author | Santoprete, M. |
| author_facet | Santoprete, M. |
| citation_txt | On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems / M. Santoprete // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 15 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show that, under some additional assumptions, compatibility in the sense of Magri implies a notion of compatibility due to Fassò and Ratiu, that we dub bi-affine compatibility. We present two proofs of this fact. The first one uses the uniqueness of the connection parallelizing all the Hamiltonian vector fields tangent to the leaves of a Lagrangian foliation. The second proof uses Darboux-Nijenhuis coordinates and symplectic connections.
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| first_indexed | 2025-12-07T15:37:53Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-147160 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:37:53Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Santoprete, M. 2019-02-13T17:58:04Z 2019-02-13T17:58:04Z 2015 On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems / M. Santoprete // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 15 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 70H06; 70G45; 37K10 DOI:10.3842/SIGMA.2015.089 https://nasplib.isofts.kiev.ua/handle/123456789/147160 Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show that, under some additional assumptions, compatibility in the sense of Magri implies a notion of compatibility due to Fassò and Ratiu, that we dub bi-affine compatibility. We present two proofs of this fact. The first one uses the uniqueness of the connection parallelizing all the Hamiltonian vector fields tangent to the leaves of a Lagrangian foliation. The second proof uses Darboux-Nijenhuis coordinates and symplectic connections. This paper is a contribution to the Special Issue on Analytical Mechanics and Dif ferential Geometry in honour
 of Sergio Benenti. The full collection is available at http://www.emis.de/journals/SIGMA/Benenti.html.
 We would like to thank one of the anonymous reviewers for suggesting to us that Theorem 2
 can be proved by using the uniqueness of the connection parallelizing all the Hamiltonian vector
 fields tangent to the leaves of a Lagrangian foliation. This work was supported by an NSERC
 Discovery Grant. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems Article published earlier |
| spellingShingle | On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems Santoprete, M. |
| title | On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems |
| title_full | On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems |
| title_fullStr | On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems |
| title_full_unstemmed | On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems |
| title_short | On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems |
| title_sort | on the relationship between two notions of compatibility for bi-hamiltonian systems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147160 |
| work_keys_str_mv | AT santopretem ontherelationshipbetweentwonotionsofcompatibilityforbihamiltoniansystems |