Linear symplectomorphism - Imported Knowlpedia Content

Definition

Let $(V,\omega_V)$ and $(W,\omega_W)$ be finite-dimensional symplectic vector spaces. A linear symplectomorphism is a bijective symplectic linear map $T:V\to W$ ; explicitly,

\omega_W(Tv,Tv')=\omega_V(v,v') \quad\text{for all }v,v'\in V.

Its inverse is automatically symplectic. When $V=W$ and $\omega_V=\omega_W=\omega$ , linear symplectomorphisms are the automorphisms of the pair $(V,\omega)$ . Equivalently, $T$ is an isomorphism of vector spaces carrying the target form exactly to the source form, with no choice of basis involved.

Automorphism group

The automorphisms of $(V,\omega)$ form the symplectic group $\operatorname{Sp}(V,\omega)$ under composition. After choosing a symplectic basis, this group is represented by matrices satisfying

A^{\mathsf T}JA=J.

Taking determinants gives $(\det A)^2=1$ ; in fact every real symplectic matrix has determinant $1$ .

Relation to general symplectic maps

A symplectic linear map between finite-dimensional spaces of equal dimension is automatically a linear symplectomorphism because it is injective. In unequal dimensions a symplectic linear map can exist only from the smaller space to the larger one and need not be onto. Thus “symplectic” and “symplectomorphism” should not be conflated when source and target dimensions may differ.

Examples and conventions

Every change from one symplectic basis of $V$ to another is a linear symplectomorphism. On $(\mathbb R^2,dx\wedge dy)$ , rotations and the shear $(x,y)\mapsto(x+y,y)$ are examples, whereas $(x,y)\mapsto(2x,2y)$ is not. In Hamiltonian mechanics, “linear canonical transformation” is a common synonym when the chosen form is the standard phase-space form.

References

Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: Chapter 2, linear symplectic geometry.
Victor Guillemin and Shlomo Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. Publisher record. Relevant: Chapter 1, the linear symplectic group.