symplectomorphism

Symplectomorphisms are the homomorphisms of symplectic manifolds.

In the context of mechanics where symplectic manifolds model phase spaces, symplectomorphisms are essentially what are called *canonical transformations*.

A **symplectomorphism** or **symplectic diffeomorphism** from a symplectic manifold $(X_1,\omega_1)$ to a symplectic manifold $(X_2,\omega_2)$ is a diffeomorphism $\phi : X \to Y$ preserving the symplectic form, i.e. such that

$\phi^* \omega_2 = \omega_1
\,.$

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 *symplectomorphism group*:

$Sympl(X, \omega)
\hookrightarrow
Diff(X)
\,.$

Its Lie algebra

$\mathfrak{SymplVect}(X, \omega) \hookrightarrow \mathfrak{Vect}(X)$

is that of symplectic vector fields: those vector fields $v \in \mathfrak{Vect}(X)$ such that their Lie derivative annihilates the symplectic form

$\mathcal{L}_v \omega = 0
\,.$

The further subgroup corresponding to those symplectic vector fields which are flows of Hamiltonian vector fields coming from a smooth family of Hamiltonians

$\mathfrak{HamVect}(X, \omega) \hookrightarrow \mathfrak{SymplVect}(X, \omega) \hookrightarrow \mathfrak{Vect}(X)$

is the group of **Hamiltonian symplectomorphisms** or **Hamiltonian diffeomorphisms**.

$HamSympl(X,\omega) \hookrightarrow Sympl(X, \omega) \hookrightarrow Diff(X)
\,.$

In the generalization to n-plectic geometry there are accordingly *$n$-plectomorphisms*. See at *higher symplectic geometry*.

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.

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 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.

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.

**higher and integrated Kostant-Souriau extensions**:

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $\mathbb{G}$-principal ∞-connection)

$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$

(extension are listed for sufficiently connected $X$)

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 *Hamiltonian matrices?*.

The following example, due to Andreas Blass and Stephen Schanuel, is a 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

$vol(B_n) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}$

where $\Gamma$ is the Euler Gamma function. In dimension $2 n$, this gives

$vol(B_{2 n}) = \frac{\pi^n}{n!}$

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 permuting coordinates. The volume of the orbit space $B_2^n/S_n$ is clearly $\pi^n/n!$.

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

$(\sqrt{r_1^2 - r_2^2}e^{i\theta_1}, \sqrt{r_2^2 - r_3^2}e^{i(\theta_1 + \theta_2)}, \ldots, \sqrt{r_{n-1}^2-r_n^2}e^{i(\theta_1 + \theta_2 + \ldots + \theta_{n-1})}, r_n e^{i(\theta_1 + \theta_2 + \ldots + \theta_n)})$

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 $1 \leq j \leq n-1$.

In other words, writing $z_j = x_j + i y_j$ the symplectic 2-form

$\sum_{j=1}^n d x_j \wedge d y_j = \sum_{j=1}^n r_j d r_j \wedge d\theta_j$

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).

Lecture notes include

- Augustin Banyaga,
*Introduction to the geometry of hamiltonian diffeomorphisms*(pdf)

The example of volumes of balls is discussed in

- Andreas Blass, Stephen Schanuel, On the volumes of balls (ps).

Revised on January 8, 2015 21:59:20
by Urs Schreiber
(195.113.30.252)