Editing Symplectomorphism (section)

==Formal definition==

A [[diffeomorphism]] between two [[symplectic manifold]]s <math>f: (M,\omega) \rightarrow (N,\omega')</math> is called a '''symplectomorphism''' if
:<math>f^*\omega'=\omega,</math>
where <math>f^*</math> is the [[pullback (differential geometry)|pullback]] of <math>f</math>. The symplectic diffeomorphisms from <math>M</math> to <math>M</math> are a (pseudo-)group, called the symplectomorphism group (see below).

The infinitesimal version of symplectomorphisms gives the symplectic vector fields. A vector field <math>X \in \Gamma^{\infty}(TM)</math> is called symplectic if
:<math>\mathcal{L}_X\omega=0.</math>
Also, <math>X</math> is symplectic if the flow <math>\phi_t: M\rightarrow M</math> of <math>X</math> is a symplectomorphism for every <math>t</math>.
These vector fields build a Lie subalgebra of <math>\Gamma^{\infty}(TM)</math>.
Here, <math>\Gamma^{\infty}(TM)</math> is the set of [[smooth function|smooth]] [[vector field]]s on <math>M</math>, and <math>\mathcal{L}_X</math> is the [[Lie derivative]] along the vector field <math>X.</math>

Examples of symplectomorphisms include the [[canonical transformation]]s of [[classical mechanics]] and [[theoretical physics]], the flow associated to any Hamiltonian function, the map on [[cotangent bundle]]s induced by any diffeomorphism of manifolds, and the coadjoint action of an element of a [[Lie group]] on a [[coadjoint orbit]].