On the Darboux theorem for weak symplectic manifolds
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- by Dario Bambusi PDF
- Proc. Amer. Math. Soc. 127 (1999), 3383-3391 Request permission
Abstract:
A new tool to study reducibility of a weak symplectic form to a constant one is introduced and used to prove a version of the Darboux theorem more general than previous ones. More precisely, at each point of the considered manifold a Banach space is associated to the symplectic form (dual of the phase space with respect to the symplectic form), and it is shown that the Darboux theorem holds if such a space is locally constant. The following application is given. Consider a weak symplectic manifold $M$ on which the Darboux theorem is assumed to hold (e.g. a symplectic vector space). It is proved that the Darboux theorem holds also for any finite codimension symplectic submanifolds of $M$, and for symplectic manifolds obtained from $M$ by the Marsden–Weinstein reduction procedure.References
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Additional Information
- Dario Bambusi
- Affiliation: Dipartimento di Matematica dell’Università, Via Saldini 50, 20133 Milano, Italy
- MR Author ID: 239364
- Email: bambusi@mat.unimi.it
- Received by editor(s): December 1, 1997
- Received by editor(s) in revised form: January 21, 1998
- Published electronically: May 3, 1999
- Additional Notes: This work was supported by grants CE n. CHRX–CT93–0330/DG, “Order and chaos in conservative dynamical systems”, and CE. n. ERBCHRXCT940460 “Stability and universality in classical mechanics".
- Communicated by: Peter Li
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 3383-3391
- MSC (1991): Primary 58B20; Secondary 58F05
- DOI: https://doi.org/10.1090/S0002-9939-99-04866-2
- MathSciNet review: 1605923