98-18 Bambusi, Dario

On Darboux theorem for weak symplectic manifolds (382K, PS) Jan 14, 98

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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 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 Darboux theorem holds if such a space is locally constant. The following application is given. Consider a weak symplectic manifold $M$ on which Darboux theorem is assumed to hold (e.g. a symplectic vector space). It is proved that Darboux theorem holds also for any finite codimension symplectic submanifolds of $M$, and for symplectic manifolds obtained from $M$ by Marsden--Weinstein reduction procedure.

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