Abstract mechanical connection and abelian reconstruction for almost Kähler manifolds
When the phase space P of a Hamiltonian G-system (P,ω,G,J,H) has an almost Kähler structure a preferred connection, called abstract mechanical connection, can be defined by declaring horizontal spaces at each point to be metric orthogonal to the tangent to the group orbit. Explicit formulas for...
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2001-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/S1110757X01000043 |
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| _version_ | 1832555080997404672 |
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| author | Sergey Pekarsky Jerrold E. Marsden |
| author_facet | Sergey Pekarsky Jerrold E. Marsden |
| author_sort | Sergey Pekarsky |
| collection | DOAJ |
| description | When the phase space P
of a Hamiltonian
G-system (P,ω,G,J,H)
has an almost Kähler structure a preferred
connection, called abstract mechanical connection, can
be defined by declaring horizontal spaces at each point to be
metric orthogonal to the tangent to the group orbit. Explicit
formulas for the corresponding connection one-form
𝒜 are derived in terms of the momentum map,
symplectic and complex structures. Such connection can play the
role of the reconstruction connection (due to the work of A.
Blaom), thus
significantly simplifying computations of the corresponding
dynamic and geometric phases for an Abelian group G. These
ideas are illustrated using the example of the resonant
three-wave interaction. Explicit formulas for the connection
one-form and the phases are given together with some new results
on the symmetry reduction of the Poisson structure. |
| format | Article |
| id | doaj-art-ddb28afaff3547c48e470e485c0b9b9a |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2001-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-ddb28afaff3547c48e470e485c0b9b9a2025-02-03T05:49:42ZengWileyJournal of Applied Mathematics1110-757X1687-00422001-01-011112810.1155/S1110757X01000043Abstract mechanical connection and abelian reconstruction for almost Kähler manifoldsSergey Pekarsky0Jerrold E. Marsden1Control and Dynamical Systems, 107-81, California Institute of Technology, Pasadena 91125, CA, USAControl and Dynamical Systems, 107-81, California Institute of Technology, Pasadena 91125, CA, USAWhen the phase space P of a Hamiltonian G-system (P,ω,G,J,H) has an almost Kähler structure a preferred connection, called abstract mechanical connection, can be defined by declaring horizontal spaces at each point to be metric orthogonal to the tangent to the group orbit. Explicit formulas for the corresponding connection one-form 𝒜 are derived in terms of the momentum map, symplectic and complex structures. Such connection can play the role of the reconstruction connection (due to the work of A. Blaom), thus significantly simplifying computations of the corresponding dynamic and geometric phases for an Abelian group G. These ideas are illustrated using the example of the resonant three-wave interaction. Explicit formulas for the connection one-form and the phases are given together with some new results on the symmetry reduction of the Poisson structure.http://dx.doi.org/10.1155/S1110757X01000043 |
| spellingShingle | Sergey Pekarsky Jerrold E. Marsden Abstract mechanical connection and abelian reconstruction for almost Kähler manifolds Journal of Applied Mathematics |
| title | Abstract mechanical connection and abelian reconstruction for
almost Kähler manifolds |
| title_full | Abstract mechanical connection and abelian reconstruction for
almost Kähler manifolds |
| title_fullStr | Abstract mechanical connection and abelian reconstruction for
almost Kähler manifolds |
| title_full_unstemmed | Abstract mechanical connection and abelian reconstruction for
almost Kähler manifolds |
| title_short | Abstract mechanical connection and abelian reconstruction for
almost Kähler manifolds |
| title_sort | abstract mechanical connection and abelian reconstruction for almost kahler manifolds |
| url | http://dx.doi.org/10.1155/S1110757X01000043 |
| work_keys_str_mv | AT sergeypekarsky abstractmechanicalconnectionandabelianreconstructionforalmostkahlermanifolds AT jerroldemarsden abstractmechanicalconnectionandabelianreconstructionforalmostkahlermanifolds |