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symplectic complement

(Definition)

Definition [1,2] Let $(V,\omega)$ be a symplectic vector space and let be a vector subspace of . Then the symplectic complement of is

$\displaystyle W^\omega = \{x\in V\, \vert \, \omega(x,y)=0\,\,$ for all $\displaystyle \,\, y\in W\}.$

It is easy to see that $W^\omega$ is also a vector subspace of . Depending on the relation between and $W^\omega$ , is given different names.

For the symplectic complement, we have the following dimension theorem.

Theorem [1,2] Let $(V,\omega)$ be a symplectic vector space, and let be a vector subspace of . Then

$\displaystyle \dim V = \dim W^\omega + \dim W.$

Bibliography

1: D. McDuff, D. Salamon, Introduction to Symplectic Topology, Clarendon Press, 1997.
2: R. Abraham, J.E. Marsden, Foundations of Mechanics, 2nd ed., Perseus Books, 1978.

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Also defines:

symplectic complement, isotropic subspace, coisotropic subspace, symplectic subspace, Lagrangian subspace

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Attachments:: dimension theorem for symplectic complement (proof) (Proof) by matte

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Cross-references: dimension, relation, easy to see, vector subspace, symplectic vector space
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This is version 5 of symplectic complement, born on 2003-04-02, modified 2004-02-28.
Object id is 4139, canonical name is SymplecticComplement.
Accessed 5477 times total.

Classification:

15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)

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