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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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2015 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2015
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147160 |
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| Cite this: | 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| id |
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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 |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems |
| spellingShingle |
On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems Santoprete, M. |
| title_short |
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_sort |
on the relationship between two notions of compatibility for bi-hamiltonian systems |
| author |
Santoprete, M. |
| author_facet |
Santoprete, M. |
| publishDate |
2015 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| 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.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147160 |
| 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. |
| work_keys_str_mv |
AT santopretem ontherelationshipbetweentwonotionsofcompatibilityforbihamiltoniansystems |
| first_indexed |
2025-12-07T15:37:53Z |
| last_indexed |
2025-12-07T15:37:53Z |
| _version_ |
1850864433707876352 |