| Version 5 |
Version 4 |
| Symplectic manifolds constitute |
Symplectic manifolds constitute |
| the mathematical structure for modern Hamiltonian mechanics. |
the mathematical structure for modern Hamiltonian mechanics. |
| Symplectic manifolds can also be seen as even dimensional |
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| analogues to contact manifolds. |
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| \begin{defn} |
\begin{defn} |
| A {\em symplectic manifold} is a pair $(M,\omega)$ consisting |
A {\em symplectic manifold} is a pair $(M,\omega)$ consisting |
| of a smooth manifold $M$ and a |
of a smooth manifold $M$ and a |
| closed \PMlinkname{2-form}{DifferentialForms} |
closed \PMlinkname{2-form}{DifferentialForms} |
| $\omega\in\Omega^2(M)$, that is non-degenerate |
$\omega\in\Omega^2(M)$, that is non-degenerate |
| at each point. |
at each point. |
| Then $\omega$ is called a {\em symplectic |
Then $\omega$ is called a {\em symplectic |
| form} for $M$. |
form} for $M$. |
| \end{defn} |
\end{defn} |
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| \subsubsection*{Properties} |
\subsubsection*{Properties} |
| \begin{enumerate} |
\begin{enumerate} |
| \item Every symplectic manifold is even dimensional. This is |
\item Every symplectic manifold is even dimensional. This is |
| easy to understand in view of the physics. In Hamilton |
easy to understand in view of the physics. In Hamilton |
| equations, location and momentum vectors always appear in pairs. |
equations, location and momentum vectors always appear in pairs. |
| \item a form $\omega\in \Omega^2(M)$ on a $2n$-dimensional |
\item a form $\omega\in \Omega^2(M)$ on a $2n$-dimensional |
| manifold $M$ is non-degenerate if and only if the |
manifold $M$ is non-degenerate if and only if the |
| $n$-fold product $\omega^n= \omega\wedge \cdots \wedge \omega$ |
$n$-fold product $\omega^n= \omega\wedge \cdots \wedge \omega$ |
| is non-zero. |
is non-zero. |
| \end{enumerate} |
\end{enumerate} |
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| Let $(M,\omega)$ and $(N,\eta)$ be symplectic manifolds. Then a diffeomorphism $f\colon M\to N$ is |
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 $N$ |
called a {\em symplectomorphism} if $f^*\eta=\omega$, that is, if the symplectic form on $N$ |
| pulls back to the form on $M$. |
pulls back to the form on $M$. |
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| \subsubsection*{Notes} |
\subsubsection*{Notes} |
| A symplectomorphism is also known as a \emph{canonical transformation}. |
A symplectomorphism is also known as a \emph{canonical transformation}. |
|
This term is mostly used in the mechanics literature.
|
This term is mostly used in the mechanics literature, e.g. \cite{abraham}
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