ConceptsInSymplecticGeometry 2004-10-26 02:56:35 2006-09-20 14:08:05 Topic % 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} \usepackage{amsthm} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions % % 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}} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\abs}[1]{| #1 |} \newtheorem{thm}{Theorem} \newtheorem{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{cor}{Corollary} \subsubsection*{Linear theory} \begin{enumerate} \item symplectic vector space, symplectic matrix \item symplectic complement \end{enumerate} \subsubsection*{Symplectic manifolds} \begin{enumerate} \item symplectic manifold (\PMlinkname{examples}{ExamplesOfSymplecticManifolds}) \item Lagrangian submanifold, isotropic submanifold \item symplectic vector field \item symplectomorphism, canonical transformation \item Hamilton equations \item \PMlinkname{Darboux's Theorem}{DarbouxsTheoremSymplecticGeometry} \item Poisson bracket \item Moser's theorem \item Gray stability theorem \item momentum map \end{enumerate}