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,ω 1)(X_1,\omega_1) to a symplectic manifold (X 2,ω 2)(X_2,\omega_2) is a diffeomorphism ϕ:XY\phi : X \to Y preserving the symplectic form, i.e. such that

ϕ *ω 2=ω 1. \phi^* \omega_2 = \omega_1 \,.


The symplectomorphisms from a symplectic manifold (X,ω)(X, \omega) to itself form an infinite-dimensional Lie group that is a subgroup of the diffeomorphism group of XX, the symplectomorphism group:

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

Its Lie algebra

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

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

vω=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

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

is the group of Hamiltonian symplectomorphisms or Hamiltonian diffeomorphisms.

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


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


Preservation of volume

Inasmuch as a symplectic manifold (M,ω)(M, \omega) carries a canonical volume form ω n\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,ω)(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,ω)(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.

Relation to Lagrangian correspondences

A symplectomorphisms ϕ:(X 1,ω 1)(X 2,ω 2)\phi \;\colon\; (X_1, \omega_1) \longrightarrow (X_2, \omega_2) canonically induces a Lagrangian correspondence between (X 1,ω 1)(X_1, \omega_1) and (X 2,ω 2)(X_2,\omega_2), given by its graph.

Extensions under geometric quantization

higher and integrated Kostant-Souriau extensions:

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

(Ω𝔾)FlatConn(X)QuantMorph(X,)HamSympl(X,) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
\inftyhigher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of (Ω𝔾)(\Omega \mathbb{G})-flat ∞-connections on XXquantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
nnn-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
nnsmooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected XX)


Linear symplectomorphisms

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

The linear Hamiltonian symplectomorphisms are also known as the Hamiltonian matrices?.

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 nn, the volume of the unit ball in n\mathbb{R}^n (with respect to the Lebesgue measure) is

vol(B n)=π n/2Γ(n2+1) vol(B_n) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}

where Γ\Gamma is the Euler Gamma function. In dimension 2n2 n, this gives

vol(B 2n)=π nn! vol(B_{2 n}) = \frac{\pi^n}{n!}

Meanwhile, we may regard π n\pi^n as the volume of the nn-dimensional complex polydisc, viz. the n thn^{th} cartesian power of the complex 1-disc B 2={z:|z|1}B_{2} = \{z: {|z|} \leq 1\}, on which the symmetric group S nS_n acts by permuting coordinates. The volume of the orbit space B 2 n/S nB_2^n/S_n is clearly π n/n!\pi^n/n!.

Theorem (Blass-Schanuel)

Given (z 1,,z n) n(z_1, \ldots, z_n) \in \mathbb{C}^n, write coordinates z jz_j in polar coordinate form z j=r je iθ jz_j = r_j e^{i \theta_j}, and define an S nS_n-invariant map ϕ:B 2 nB 2n\phi \colon B_2^n \to B_{2 n} by first permuting the z jz_j so that r 1r 2r nr_1 \geq r_2 \geq \ldots \geq r_n and then mapping (z 1,,z n)(z_1, \ldots, z_n) to

(r 1 2r 2 2e iθ 1,r 2 2r 3 2e i(θ 1+θ 2),,r n1 2r n 2e i(θ 1+θ 2++θ n1),r ne i(θ 1+θ 2++θ 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 ϕ\phi induces a continuous well-defined map B 2 n/S nB 2nB_2^n/S_n \to B_{2 n}. Furthermore, when restricted to the set P nP_n of (z 1,,z n)(z_1, \ldots, z_n) for which the r jr_j are all distinct, ϕ\phi induces a smooth symplectic isomorphism mapping P n/S nP_n/S_n onto the set Q nQ_n of (w 1,,w n)B 2n(w_1, \ldots, w_n) \in B_{2 n} for which w j0w_j \neq 0 for 1jn11 \leq j \leq n-1.

In other words, writing z j=x j+iy jz_j = x_j + i y_j the symplectic 2-form

j=1 ndx jdy j= j=1 nr jdr jdθ 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 ϕ:P n/S nQ n\phi \colon P_n/S_n \to Q_n. Since symplectic maps are locally volume-preserving, and since P nP_n and Q nQ_n are almost all of B 2 nB_2^n and B 2nB_{2 n} respectively, this gives a proof that the volume of B 2nB_{2 n} is π n/n!\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 (