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

DOI:
https://doi.org/10.1090/S0002-9939-99-04866-2

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. E. Marsden: Darboux's Theorem Fails for Weak Symplectic Forms. Proc. Amer. Math. Soc.**32**, 590-592 (1972). MR**45:2755****2.**A.J. Tromba: Almost Riemannian Structures on Banach Manifolds: the Morse Lemma and the Darboux Theorem. Canad. J. Math.**28**, 640-652 (1976). MR**53:6613****3.**J.E. Marsden: Lectures on Geometric Methods in Mathematical Physics. SIAM (Philadelphia, 1981). MR**82j:58046****4.**D. Bambusi, A. Giorgilli: Exponential Stability of States Close to Resonance in Infinite Dimensional Hamiltonian Systems. Jour. Stat. Phys.**71**, p. 569-606 (1993). MR**94m:58198****5.**A. Weinstein: Symplectic Structures on Banach Manifolds. Bull. Amer. Math. Soc.**75**, 1040-1041 (1969)**6.**R. Schmid: Infinite dimensional Hamiltonian systems. Bibliopolis (Napoli 1987). MR**90f:58072**

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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:
https://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