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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On the Darboux theorem
for weak symplectic manifolds

Author: Dario Bambusi
Journal: Proc. Amer. Math. Soc. 127 (1999), 3383-3391
MSC (1991): Primary 58B20; Secondary 58F05
Published electronically: May 3, 1999
MathSciNet review: 1605923
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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.

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Additional Information

Dario Bambusi
Affiliation: Dipartimento di Matematica dell’Università, Via Saldini 50, 20133 Milano, Italy

PII: S 0002-9939(99)04866-2
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
Article copyright: © Copyright 1999 American Mathematical Society

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