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\begin{document}

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\section*{symplectomorphism}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{symplectomorphisms}{Symplectomorphisms}\dotfill \pageref*{symplectomorphisms} \linebreak
\noindent\hyperlink{AutoSymplectomorphisms}{Auto-symplectomorphisms}\dotfill \pageref*{AutoSymplectomorphisms} \linebreak
\noindent\hyperlink{plectomorphisms}{$n$-Plectomorphisms}\dotfill \pageref*{plectomorphisms} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{preservation_of_volume}{Preservation of volume}\dotfill \pageref*{preservation_of_volume} \linebreak
\noindent\hyperlink{relation_to_poisson_brackets}{Relation to Poisson brackets}\dotfill \pageref*{relation_to_poisson_brackets} \linebreak
\noindent\hyperlink{relation_to_lagrangian_correspondences}{Relation to Lagrangian correspondences}\dotfill \pageref*{relation_to_lagrangian_correspondences} \linebreak
\noindent\hyperlink{extensions_under_geometric_quantization}{Extensions under geometric quantization}\dotfill \pageref*{extensions_under_geometric_quantization} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{linear_symplectomorphisms}{Linear symplectomorphisms}\dotfill \pageref*{linear_symplectomorphisms} \linebreak
\noindent\hyperlink{a_curious_example_volumes_of_balls}{A curious example: volumes of balls}\dotfill \pageref*{a_curious_example_volumes_of_balls} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Symplectomorphisms are the [[homomorphisms]] of [[symplectic manifolds]].

In the context of [[mechanics]] where symplectic manifolds model [[phase spaces]], symplectomorphisms are essentially what are called \emph{[[canonical transformations]]}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{symplectomorphisms}{}\subsubsection*{{Symplectomorphisms}}\label{symplectomorphisms}

A \textbf{symplectomorphism} or \textbf{symplectic diffeomorphism} from a [[symplectic manifold]] $(X_1,\omega_1)$ to a symplectic manifold $(X_2,\omega_2)$ is a [[diffeomorphism]] $\phi : X_1 \to X_2$ preserving the [[symplectic form]], i.e. such that

\begin{displaymath}
\phi^* \omega_2 = \omega_1
  \,.
\end{displaymath}
\hypertarget{AutoSymplectomorphisms}{}\subsubsection*{{Auto-symplectomorphisms}}\label{AutoSymplectomorphisms}

The symplectomorphisms from a [[symplectic manifold]] $(X, \omega)$ to itself form an infinite-dimensional [[Lie group]] that is a [[subgroup]] of the [[diffeomorphism group]] of $X$, the \emph{[[symplectomorphism group]]}:

\begin{displaymath}
Sympl(X, \omega) 
  \hookrightarrow
  Diff(X)
  \,.
\end{displaymath}
Its [[Lie algebra]]

\begin{displaymath}
\mathfrak{SymplVect}(X, \omega) \hookrightarrow \mathfrak{Vect}(X)
\end{displaymath}
is that of [[symplectic vector fields]]: those [[vector fields]] $v \in  \mathfrak{Vect}(X)$ such that their [[Lie derivative]] annihilates the [[symplectic form]]

\begin{displaymath}
\mathcal{L}_v \omega = 0
  \,.
\end{displaymath}
The further [[subgroup]] corresponding to those symplectic vector fields which are [[flows]] of [[Hamiltonian vector fields]] coming from a smooth family of [[Hamiltonians]]

\begin{displaymath}
\mathfrak{HamVect}(X, \omega) \hookrightarrow \mathfrak{SymplVect}(X, \omega) \hookrightarrow \mathfrak{Vect}(X)
\end{displaymath}
is the group of \textbf{Hamiltonian symplectomorphisms} or \textbf{Hamiltonian diffeomorphisms}.

\begin{displaymath}
HamSympl(X,\omega) \hookrightarrow Sympl(X, \omega) \hookrightarrow Diff(X)
  \,.
\end{displaymath}
\hypertarget{plectomorphisms}{}\subsubsection*{{$n$-Plectomorphisms}}\label{plectomorphisms}

In the generalization to [[n-plectic geometry]] there are accordingly \emph{$n$-plectomorphisms}. See at \emph{[[higher symplectic geometry]]}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{preservation_of_volume}{}\subsubsection*{{Preservation of volume}}\label{preservation_of_volume}

Inasmuch as a symplectic manifold $(M, \omega)$ carries a canonical [[volume form]] $\omega^{\wedge n}$, it is clear that a symplectomorphism is locally volume-preserving.

\hypertarget{relation_to_poisson_brackets}{}\subsubsection*{{Relation to Poisson brackets}}\label{relation_to_poisson_brackets}

The [[Lie algebra]] given by the [[Poisson bracket]] of a [[symplectic manifold]] $(X, \omega)$ is that of a [[central extension]] of the group of Hamiltonian symplectomorphisms. (It [[Lie integration|integrates]] to the [[quantomorphism group]].)

The central extension results form the fact that the Hamiltonian associated with every [[Hamiltonian vector field]] is well defined only up to the addition of a constant function.

If $(X, \omega)$ is a [[symplectic vector space]] then there is corresponding to it a [[Heisenberg Lie algebra]]. This sits inside the Poisson bracket algebra, and accordingly the [[Heisenberg group]] is a subgroup of the group of (necessarily Hamiltonian) symplectomorphisms of the symplectic vector space, regarded as a symplectic manifold.

\hypertarget{relation_to_lagrangian_correspondences}{}\subsubsection*{{Relation to Lagrangian correspondences}}\label{relation_to_lagrangian_correspondences}

A symplectomorphisms $\phi \;\colon\; (X_1, \omega_1) \longrightarrow (X_2, \omega_2)$ canonically induces a [[Lagrangian correspondence]] between $(X_1, \omega_1)$ and $(X_2,\omega_2)$, given by its [[graph]].

\hypertarget{extensions_under_geometric_quantization}{}\subsubsection*{{Extensions under geometric quantization}}\label{extensions_under_geometric_quantization}

[[!include geometric quantization extensions - table]]

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{linear_symplectomorphisms}{}\subsubsection*{{Linear symplectomorphisms}}\label{linear_symplectomorphisms}

Given a [[symplectic vector space]] $(V,\omega)$ regarded as a [[symplectic manifold]], then those symplectomorphisms which are [[linear maps]] on $V$ form, under composition, the [[symplectic group]] $Sp(V,\omega)$.

The linear Hamiltonian symplectomorphisms are also known as the \emph{[[Hamiltonian matrices]]}.

\hypertarget{a_curious_example_volumes_of_balls}{}\subsubsection*{{A curious example: volumes of balls}}\label{a_curious_example_volumes_of_balls}

The following example, due to Andreas Blass and Stephen Schanuel, is a [[categorification|categorified]] way to calculate [[volumes]] of even-dimensional [[balls]].

In any dimension $n$, the volume of the unit ball in $\mathbb{R}^n$ (with respect to the [[Lebesgue measure]]) is

\begin{displaymath}
vol(B_n) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}
\end{displaymath}
where $\Gamma$ is the Euler [[Gamma function]]. In dimension $2 n$, this gives

\begin{displaymath}
vol(B_{2 n}) = \frac{\pi^n}{n!}
\end{displaymath}
Meanwhile, we may regard $\pi^n$ as the volume of the $n$-dimensional complex [[polydisc]], viz. the $n^{th}$ cartesian power of the complex 1-disc $B_{2} = \{z: {|z|} \leq 1\}$, on which the [[symmetric group]] $S_n$ acts by [[permutation|permuting]] [[coordinates]]. The volume of the orbit space $B_2^n/S_n$ is clearly $\pi^n/n!$.

\begin{uthm}
Given $(z_1, \ldots, z_n) \in \mathbb{C}^n$, write coordinates $z_j$ in polar coordinate form $z_j = r_j e^{i \theta_j}$, and define an $S_n$-invariant map $\phi \colon B_2^n \to B_{2 n}$ by first permuting the $z_j$ so that $r_1 \geq r_2 \geq \ldots \geq r_n$ and then mapping $(z_1, \ldots, z_n)$ to

\begin{displaymath}
(\sqrt{r_1^2 - r_2^2}e^{i\theta_1}, \sqrt{r_2^2 - r_3^2}e^{i(\theta_1 + \theta_2)}, \ldots, \sqrt{r_{n-1}^2-r_n^2}e^{i(\theta_1 + \theta_2 + \ldots + \theta_{n-1})}, r_n e^{i(\theta_1 + \theta_2 + \ldots + \theta_n)})
\end{displaymath}
Then $\phi$ induces a continuous well-defined map $B_2^n/S_n \to B_{2 n}$. Furthermore, when restricted to the set $P_n$ of $(z_1, \ldots, z_n)$ for which the $r_j$ are all distinct, $\phi$ induces a smooth symplectic isomorphism mapping $P_n/S_n$ onto the set $Q_n$ of $(w_1, \ldots, w_n) \in B_{2 n}$ for which $w_j \neq 0$ for $1 \leq j \leq n-1$.

\end{uthm}
In other words, writing $z_j = x_j + i y_j$ the symplectic 2-form

\begin{displaymath}
\sum_{j=1}^n d x_j \wedge d y_j = \sum_{j=1}^n r_j d r_j \wedge d\theta_j
\end{displaymath}
is preserved by pulling back along $\phi \colon P_n/S_n \to Q_n$. Since symplectic maps are locally volume-preserving, and since $P_n$ and $Q_n$ are almost all of $B_2^n$ and $B_{2 n}$ respectively, this gives a proof that the volume of $B_{2 n}$ is $\pi^n/n!$ (alternate to standard purely computational proofs).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[canonical transformation]]


\item [[Hamilton's equations]]


\item [[symplectic integrator]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Lecture notes include

\begin{itemize}%
\item Augustin Banyaga, \emph{Introduction to the geometry of hamiltonian diffeomorphisms} (\href{http://www.math.psu.edu/wade/dakar.pdf}{pdf})

\end{itemize}
The example of volumes of balls is discussed in

\begin{itemize}%
\item Andreas Blass, Stephen Schanuel, On the volumes of balls (\href{http://www.math.lsa.umich.edu/~ablass/vol.ps}{ps}).

\end{itemize}
[[!redirects symplectomorphisms]]

[[!redirects Hamiltonian symplectomorphism]] [[!redirects Hamiltonian symplectomorphisms]] [[!redirects hamiltonian symplectomorphism]] [[!redirects hamiltonian symplectomorphisms]]

[[!redirects n-plectomorphism]] [[!redirects n-plectomorphisms]] [[!redirects Hamiltonian n-plectomorphism]] [[!redirects Hamiltonian n-plectomorphisms]]

[[!redirects symplectic diffeomorphism]] [[!redirects symplectic diffeomorphisms]] [[!redirects Hamiltonian diffeomorphism]] [[!redirects Hamiltonian diffeomorphisms]]



\end{document}