symplectic manifold SymplecticManifold 2002-12-05 23:30:37 2006-07-09 12:21:59 Definition symplectic form symplectomorphism canonical transformation % 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} Symplectic manifolds constitute the mathematical structure for modern Hamiltonian mechanics. Symplectic manifolds can also be seen as even dimensional analogues to contact manifolds. \begin{defn} A {\em symplectic manifold} is a pair $(M,\omega)$ consisting of a smooth manifold $M$ and a closed \PMlinkname{2-form}{DifferentialForms} $\omega\in\Omega^2(M)$, that is non-degenerate at each point. Then $\omega$ is called a {\em symplectic form} for $M$. \end{defn} \subsubsection*{Properties} \begin{enumerate} \item Every symplectic manifold is even dimensional. This is easy to understand in view of the physics. In Hamilton equations, location and momentum vectors always appear in pairs. \item A form $\omega\in \Omega^2(M)$ on a $2n$-dimensional manifold $M$ is non-degenerate if and only if the $n$-fold product $\omega^n= \omega\wedge \cdots \wedge \omega$ is non-zero. \item As a consequence of the last \PMlinkescapetext{property}, every symplectic manifold is orientable. \end{enumerate} Let $(M,\omega)$ and $(N,\eta)$ be symplectic manifolds. Then a diffeomorphism $f\colon M\to N$ is called a {\em symplectomorphism} if $f^*\eta=\omega$, that is, if the symplectic form on $N$ pulls back to the form on $M$. \subsubsection*{Notes} A symplectomorphism is also known as a \emph{canonical transformation}. This \PMlinkescapetext{term} is mostly used in the mechanics literature.