Symplectomorphism - Knowlpedia

Let $(M,\omega_M)$ and $(N,\omega_N)$ be symplectic manifolds. A symplectomorphism from $M$ to $N$ is a diffeomorphism $\varphi:M\to N$ satisfying

\varphi^*\omega_N=\omega_M,

where $\varphi^*$ denotes pullback of differential forms. Equivalently, $\varphi$ is an invertible symplectic map. Pointwise, its differential is a symplectic linear isomorphism:

\omega_N(d\varphi_pu,d\varphi_pv)=\omega_M(u,v)

for every $p\in M$ and $u,v\in T_pM$ . Thus a symplectomorphism preserves the full symplectic form, not merely the topology, orientation, or associated volume.

Group and local structure

The identity is a symplectomorphism, composites of symplectomorphisms are symplectomorphisms, and the inverse of a symplectomorphism is symplectic. Hence the self-symplectomorphisms of $(M,\omega)$ form the symplectomorphism group $\operatorname{Symp}(M,\omega)$ . They are also the isomorphisms in the category of symplectic manifolds. The symplectic Darboux theorem says every two symplectic manifolds of the same dimension are locally symplectomorphic, although global symplectomorphism can be obstructed.

Dynamics

Whenever a Hamiltonian vector field has a flow defined through time $t$ , its time- $t$ map is a symplectomorphism. More generally, a vector field $X$ generates local symplectomorphisms exactly when $\mathcal L_X\omega=0$ . Hamiltonian flows are a distinguished subclass because $\iota_X\omega$ is exact, not merely closed.

A path of self-symplectomorphisms beginning at the identity is a symplectic isotopy. Its endpoint lies in $\operatorname{Symp}_0(M,\omega)$ ; requiring an exact generating contraction form gives a Hamiltonian isotopy.

Examples and non-examples

Every linear map $A:\mathbb R^{2n}\to\mathbb R^{2n}$ satisfying $A^{\mathsf T}JA=J$ is a symplectomorphism for the standard form. A dilation $x\mapsto cx$ pulls the standard form back to $c^2\omega$ , so it is symplectic only when $c^2=1$ . An orientation-preserving, volume-preserving diffeomorphism in dimension greater than two need not preserve the symplectic form.

Conventions and scope

In Hamiltonian mechanics, “canonical transformation” commonly means symplectomorphism, though conventions involving time-dependent or contact transformations may broaden the term.

References

Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: §1.1, symplectic manifolds, maps, and diffeomorphisms.
Vladimir I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989. DOI record. Relevant: chapters on symplectic geometry and canonical transformations.