symplectic complement

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

Wω={xV|ω(x,y)=0for allyW}.

It is easy to see that Wω is also a vector subspace of V. Depending on the relationMathworldPlanetmath between W and Wω, W is given different names.

  1. 1.

    If WWω, then W is an isotropic subspace (of V).

  2. 2.

    If WωW, then W is an coisotropic subspace.

  3. 3.

    If WWω={0}, then W is an symplectic subspace.

  4. 4.

    If W=Wω, then W is an Lagrangian subspace.

For the symplectic complement, we have the following dimensionPlanetmathPlanetmath theoremMathworldPlanetmath.

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



  • 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.
Title symplectic complement
Canonical name SymplecticComplement
Date of creation 2013-03-22 13:32:25
Last modified on 2013-03-22 13:32:25
Owner matte (1858)
Last modified by matte (1858)
Numerical id 8
Author matte (1858)
Entry type Definition
Classification msc 15A04
Defines symplectic complement
Defines isotropic subspace
Defines coisotropic subspace
Defines symplectic subspace
Defines Lagrangian subspace