Real Forms of Holomorphic Hamiltonian Systems
By complexifying a Hamiltonian system, one obtains dynamics on a holomorphic symplectic manifold. To invert this construction, we present a theory of real forms that not only recovers the original system but also yields different real Hamiltonian systems that share the same complexification. This pr...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212778 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Real Forms of Holomorphic Hamiltonian Systems. Philip Arathoon and Marine Fontaine. SIGMA 20 (2024), 114, 24 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862658065598251008 |
|---|---|
| author | Arathoon, Philip Fontaine, Marine |
| author_facet | Arathoon, Philip Fontaine, Marine |
| citation_txt | Real Forms of Holomorphic Hamiltonian Systems. Philip Arathoon and Marine Fontaine. SIGMA 20 (2024), 114, 24 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | By complexifying a Hamiltonian system, one obtains dynamics on a holomorphic symplectic manifold. To invert this construction, we present a theory of real forms that not only recovers the original system but also yields different real Hamiltonian systems that share the same complexification. This provides a notion of real forms for holomorphic Hamiltonian systems analogous to that of real forms for complex Lie algebras. Our main result is that the complexification of any analytic mechanical system on a Grassmannian admits a real form on a compact symplectic manifold. This produces a 'unitary trick' for Hamiltonian systems, which curiously requires an essential use of hyperkähler geometry. We demonstrate this result by finding compact real forms for the simple pendulum, the spherical pendulum, and the rigid body.
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| first_indexed | 2026-03-15T23:36:29Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212778 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T23:36:29Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Arathoon, Philip Fontaine, Marine 2026-02-11T10:37:23Z 2024 Real Forms of Holomorphic Hamiltonian Systems. Philip Arathoon and Marine Fontaine. SIGMA 20 (2024), 114, 24 pages 1815-0659 2020 Mathematics Subject Classification: 53D20; 14J42 arXiv:2009.10417 https://nasplib.isofts.kiev.ua/handle/123456789/212778 https://doi.org/10.3842/SIGMA.2024.114 By complexifying a Hamiltonian system, one obtains dynamics on a holomorphic symplectic manifold. To invert this construction, we present a theory of real forms that not only recovers the original system but also yields different real Hamiltonian systems that share the same complexification. This provides a notion of real forms for holomorphic Hamiltonian systems analogous to that of real forms for complex Lie algebras. Our main result is that the complexification of any analytic mechanical system on a Grassmannian admits a real form on a compact symplectic manifold. This produces a 'unitary trick' for Hamiltonian systems, which curiously requires an essential use of hyperkähler geometry. We demonstrate this result by finding compact real forms for the simple pendulum, the spherical pendulum, and the rigid body. At the time of writing, Philip Arathoon was funded by an EPSRC Doctoral Prize Award hosted by the University of Manchester, and Marine Fontaine was supported by the FWO-EoS Project G0H4518N. We would like to extend special thanks to James Montaldi for carefully reading an early draft and for suggesting many helpful comments throughout its preparation. Additional thanks are owed to the referees whose insights and recommendations helped to improve this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Real Forms of Holomorphic Hamiltonian Systems Article published earlier |
| spellingShingle | Real Forms of Holomorphic Hamiltonian Systems Arathoon, Philip Fontaine, Marine |
| title | Real Forms of Holomorphic Hamiltonian Systems |
| title_full | Real Forms of Holomorphic Hamiltonian Systems |
| title_fullStr | Real Forms of Holomorphic Hamiltonian Systems |
| title_full_unstemmed | Real Forms of Holomorphic Hamiltonian Systems |
| title_short | Real Forms of Holomorphic Hamiltonian Systems |
| title_sort | real forms of holomorphic hamiltonian systems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212778 |
| work_keys_str_mv | AT arathoonphilip realformsofholomorphichamiltoniansystems AT fontainemarine realformsofholomorphichamiltoniansystems |