| Version 4 |
Version 3 |
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{\bf Definition} \cite{mcduff, abraham}
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{\bf Definition} \cite{mcduff}
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| Let $(V,\omega)$ be a symplectic vector space and let $W$ be a |
Let $(V,\omega)$ be a symplectic vector space and let $W$ be a |
| vector subspace of $V$. Then the \emph{symplectic complement} of $W$ |
vector subspace of $V$. Then the \emph{symplectic complement} of $W$ |
| $$W^\omega = \{x\in V\, | \, \omega(x,y)=0\,\, \mbox{for all}\,\, y\in W\}.$$ |
$$W^\omega = \{x\in V\, | \, \omega(x,y)=0\,\, \mbox{for all}\,\, y\in W\}.$$ |
| It is easy to see that $W^\omega$ is also a vector subspace of $V$. |
It is easy to see that $W^\omega$ is also a vector subspace of $V$. |
| Depending on the relation between $W$ and $W^\omega$, |
Depending on the relation between $W$ and $W^\omega$, |
| $W$ is given different names. |
$W$ is given different names. |
| \begin{enumerate} |
\begin{enumerate} |
| \item If $W\subset W^\omega$, then $W$ is an \emph{isotropic subspace} (of $V$). |
\item If $W\subset W^\omega$, then $W$ is an \emph{isotropic subspace} (of $V$). |
| \item If $W^\omega \subset W$, then $W$ is an \emph{coisotropic subspace}. |
\item If $W^\omega \subset W$, then $W$ is an \emph{coisotropic subspace}. |
| \item If $W \cap W^\omega=\{0\}$, then $W$ is an \emph{symplectic subspace}. |
\item If $W \cap W^\omega=\{0\}$, then $W$ is an \emph{symplectic subspace}. |
| \item If $W = W^\omega$, then $W$ is an \emph{Lagrangian subspace}. |
\item If $W = W^\omega$, then $W$ is an \emph{Lagrangian subspace}. |
| \end{enumerate} |
\end{enumerate} |
| For the symplectic complement, we have the |
For the symplectic complement, we have the |
| following dimension theorem. |
following dimension theorem. |
|
{\bf Theorem} \cite{mcduff, abraham} Let $(V,\omega)$ be a symplectic vector
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{\bf Theorem} \cite{mcduff} Let $(V,\omega)$ be a symplectic vector
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| space, and let $W$ be a vector subspace of $V$. Then |
space, and let $W$ be a vector subspace of $V$. Then |
| $$\dim V = \dim W^\omega + \dim W.$$ |
$$\dim V = \dim W^\omega + \dim W.$$ |
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem {mcduff} D. McDuff, D. Salamon, |
\bibitem {mcduff} D. McDuff, D. Salamon, |
| \emph{Introduction to Symplectic Topology}, |
\emph{Introduction to Symplectic Topology}, |
| Clarendon Press, 1997. |
Clarendon Press, 1997. |
| \bibitem{abraham} R. Abraham, J.E. Marsden, \emph{Foundations of Mechanics}, |
|
| 2nd ed., Perseus Books, 1978. |
|
| \end{thebibliography} |
\end{thebibliography} |
