SymplecticComplement 2003-04-02 14:16:28 2004-02-28 01:24:49 Definition symplectic vector space symplectic complement isotropic subspace coisotropic subspace symplectic subspace Lagrangian subspace % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} {\bf Definition} \cite{mcduff, abraham} Let $(V,\omega)$ be a symplectic vector space and let $W$ be a vector subspace of $V$. Then the \emph{symplectic complement} of $W$ is $$W^\omega = \{x\in V\, | \, \omega(x,y)=0\,\, \mbox{for all}\,\, y\in W\}.$$ It is easy to see that $W^\omega$ is also a vector subspace of $V$. Depending on the relation between $W$ and $W^\omega$, $W$ is given different names. \begin{enumerate} \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 $W \cap W^\omega=\{0\}$, then $W$ is an \emph{symplectic subspace}. \item If $W = W^\omega$, then $W$ is an \emph{Lagrangian subspace}. \end{enumerate} For the symplectic complement, we have the following dimension theorem. {\bf Theorem} \cite{mcduff, abraham} Let $(V,\omega)$ be a symplectic vector space, and let $W$ be a vector subspace of $V$. Then $$\dim V = \dim W^\omega + \dim W.$$ \begin{thebibliography}{9} \bibitem {mcduff} D. McDuff, D. Salamon, \emph{Introduction to Symplectic Topology}, Clarendon Press, 1997. \bibitem{abraham} R. Abraham, J.E. Marsden, \emph{Foundations of Mechanics}, 2nd ed., Perseus Books, 1978. \end{thebibliography}