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'symplectic manifold'
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| Title of object: |
symplectic manifold |
| Canonical Name: |
SymplecticManifold |
| Type: |
Definition |
| Created on: |
2002-12-05 23:30:37 |
| Modified on: |
2006-07-09 11:54:53 |
| Classification: |
msc:53D05 |
| Defines: |
symplectic form, symplectomorphism, canonical transformation |
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Content:
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 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 term is mostly used in the mechanics literature. |
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