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 | References | Similar Articles | Additional Information

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 Marsden-Weinstein reduction procedure.

**1.**J. Marsden,*Darboux’s theorem fails for weak symplectic forms*, Proc. Amer. Math. Soc.**32**(1972), 590–592. MR**0293678**, 10.1090/S0002-9939-1972-0293678-X**2.**A. J. Tromba,*Almost-Riemannian structures on Banach manifolds: the Morse lemma and the Darboux theorem*, Canad. J. Math.**28**(1976), no. 3, 640–652. MR**0402799****3.**Jerrold E. Marsden,*Lectures on geometric methods in mathematical physics*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 37, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. MR**619693****4.**Dario Bambusi and Antonio Giorgilli,*Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems*, J. Statist. Phys.**71**(1993), no. 3-4, 569–606. MR**1219023**, 10.1007/BF01058438**5.**A. Weinstein: Symplectic Structures on Banach Manifolds. Bull. Amer. Math. Soc.**75**, 1040-1041 (1969)**6.**Rudolf Schmid,*Infinite-dimensional Hamiltonian systems*, Monographs and Textbooks in Physical Science. Lecture Notes, vol. 3, Bibliopolis, Naples, 1987. MR**969603**

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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/S0002-9939-99-04866-2

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