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symplectomorphism

Idea

Symplectomorphisms are the homomorphisms of symplectic manifolds.

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

Definition

Symplectomorphisms

A symplectomorphism or symplectic diffeomorphism from a symplectic manifold $(X_1,{\omega }_1)$ to a symplectic manifold $(X_2,{\omega }_2)$ is a diffeomorphism $\varphi :X\to Y$ preserving the symplectic form, i.e. such that

$${\varphi }^{*}{\omega }_2={\omega }_1.$$\phi^* \omega_2 = \omega_1 \,.

Auto-symplectomorphisms

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:

$$\mathrm{Sympl}(X,\omega )\hookrightarrow \mathrm{Diff}(X).$$Sympl(X, \omega) \hookrightarrow Diff(X) \,.

Its Lie algebra

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

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

$${ℒ}_v\omega =0.$$\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

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

is the group of Hamiltonian symplectomorphisms or Hamiltonian diffeomorphisms.

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

$n$-Plectomorphisms

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

Properties

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.

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 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.

Extensions under geometric quantization

higher and integrated Kostant-Souriau extensions

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

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

$n$	geometry	structure	unextended structure	extension by	quantum extension
$\mathrm{\infty }$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega 𝔾)$-flat ∞-connections on $X$	quantomorphism ∞-group
1	symplectic geometry	Lie algebra	Hamiltonian vector fields	real numbers	Hamiltonians under Poisson bracket
1		Lie group	Hamiltonian symplectomorphism group	circle group	quantomorphism group
2	2-plectic geometry	Lie 2-algebra	Hamiltonian vector fields	line Lie 2-algebra	Poisson Lie 2-algebra
2		Lie 2-group	Hamiltonian 2-plectomorphisms	circle 2-group	quantomorphism 2-group
$n$	n-plectic geometry	Lie n-algebra	Hamiltonian vector fields	line Lie n-algebra	Poisson Lie n-algebra
$n$		smooth n-group	Hamiltonian n-plectomorphisms	circle n-group	quantomorphism n-group

(extension are listed for sufficiently connected $X$)

Examples

A curious example: volumes of balls

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 ${ℝ}^n$ (with respect to the Lebesgue measure) is

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

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

$$\mathrm{vol}(B_{2n})=\frac{{\pi }^n}{n!}$$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 $n^{\mathrm{th}}$ cartesian power of the complex 1-disc $B_2=\{z:\mid z\mid \le 1\}$, on which the symmetric group $S_n$ acts by permuting coordinates. The volume of the orbit space $B_2^n/S_n$ is clearly ${\pi }^n/n!$.

Theorem (Blass-Schanuel)

Given $(z_1,\dots ,z_n)\in {ℂ}^n$, write coordinates $z_j$ in polar coordinate form $z_j=r_j{e}^{i{\theta }_j}$, and define an $S_n$-invariant map $\varphi :B_2^n\to B_{2n}$ by first permuting the $z_j$ so that $r_1\ge r_2\ge \dots \ge r_n$ and then mapping $(z_1,\dots ,z_n)$ to

$$(\sqrt{r_1^2-r_2^2}{e}^{i{\theta }_1},\sqrt{r_2^2-r_3^2}{e}^{i({\theta }_1+{\theta }_2)},\dots ,\sqrt{r_{n-1}^2-r_n^2}{e}^{i({\theta }_1+{\theta }_2+\dots +{\theta }_{n-1})},r_n{e}^{i({\theta }_1+{\theta }_2+\dots +{\theta }_n)})$$(\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 $\varphi$ induces a continuous well-defined map $B_2^n/S_n\to B_{2n}$. Furthermore, when restricted to the set $P_n$ of $(z_1,\dots ,z_n)$ for which the $r_j$ are all distinct, $\varphi$ induces a smooth symplectic isomorphism mapping $P_n/S_n$ onto the set $Q_n$ of $(w_1,\dots ,w_n)\in B_{2n}$ for which $w_j\ne 0$ for $1\le j\le n-1$.

In other words, writing $z_j=x_j+i{y}_j$ the symplectic 2-form

$$\sum _{j=1}^{n}d{x}_j\wedge d{y}_j=\sum _{j=1}^n{r}_{j}d{r}_j\wedge d{\theta }_j$$\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 $\varphi :P_n/S_n\to Q_n$. Since symplectic maps are locally volume-preserving, and since $P_n$ and $Q_n$ are almost all of $B_2^n$ and $B_{2n}$ respectively, this gives a proof that the volume of $B_{2n}$ is ${\pi }^n/n!$ (alternate to standard purely computational proofs).

References

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).