On the Darboux theorem for weak symplectic manifolds
Author:
Dario Bambusi
Journal:
Proc. Amer. Math. Soc. 127 (1999), 33833391
MSC (1991):
Primary 58B20; Secondary 58F05
Published electronically:
May 3, 1999
MathSciNet review:
1605923
Fulltext PDF Free Access
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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 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 , and for symplectic manifolds obtained from by the MarsdenWeinstein reduction procedure.
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Additional Information
Dario Bambusi
Affiliation:
Dipartimento di Matematica dell’Università, Via Saldini 50, 20133 Milano, Italy
Email:
bambusi@mat.unimi.it
DOI:
http://dx.doi.org/10.1090/S0002993999048662
PII:
S 00029939(99)048662
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
