Editing Symplectic manifold (section)

== Motivation ==
Symplectic manifolds arise from [[classical mechanics]]; in particular, they are a generalization of the [[phase space]] of a closed system.<ref name="Webster">{{cite web |first=Ben |last=Webster |title=What is a symplectic manifold, really? |date=9 January 2012 |url=https://sbseminar.wordpress.com/2012/01/09/what-is-a-symplectic-manifold-really/ }}</ref> In the same way the [[Hamilton equations]] allow one to derive the time evolution of a system from a set of [[differential equation]]s, the symplectic form should allow one to obtain a [[vector field]] describing the flow of the system from the differential <math>dH</math> of a Hamiltonian function <math>H</math>.<ref name="Cohn">{{cite web |first=Henry |last=Cohn |title=Why symplectic geometry is the natural setting for classical mechanics |url=https://math.mit.edu/~cohn/Thoughts/symplectic.html }}</ref> So we require a linear map <math>TM \rightarrow T^*M </math> from the [[tangent manifold]] <math>TM</math> to the [[cotangent manifold]] <math> T^* M </math>, or equivalently, an element of <math>T^*M \otimes T^*M</math>. Letting <math>\omega</math> denote a [[Section (fiber bundle)|section]] of <math>T^*M \otimes T^* M</math>, the requirement that <math>\omega</math> be [[Degenerate form|non-degenerate]] ensures that for every differential <math>dH</math> there is a unique corresponding vector field <math>V_H</math> such that <math>dH = \omega (V_H, \cdot)</math>. Since one desires the Hamiltonian to be constant along flow lines, one should have <math>\omega(V_H, V_H) = dH(V_H) = 0</math>, which implies that <math>\omega</math> is [[Alternating form|alternating]] and hence a 2-form. Finally, one makes the requirement that <math>\omega</math> should not change under flow lines, i.e. that the [[Lie derivative]] of <math>\omega</math> along <math>V_H</math> vanishes. Applying [[Cartan homotopy formula|Cartan's formula]], this amounts to (here <math> \iota_X</math> is the [[interior product]]): 

:<math>\mathcal{L}_{V_H}(\omega) = 0\;\Leftrightarrow\;\mathrm d (\iota_{V_H} \omega) + \iota_{V_H} \mathrm d\omega= \mathrm d (\mathrm d\,H) + \mathrm d\omega(V_H) = \mathrm d\omega(V_H)=0</math>

so that, on repeating this argument for different smooth functions <math>H</math> such that the corresponding <math>V_H</math> span the tangent space at each point the argument is applied at, we see that the requirement for the vanishing Lie derivative along flows of <math>V_H</math> corresponding to arbitrary smooth <math>H</math> is equivalent to the requirement that ''ω'' should be [[Closed and exact differential forms|closed]].