proof of Darboux’s theorem (symplectic geometry)
We first observe that it suffices to prove the theorem for symplectic forms defined on an open neighbourhood of .
Indeed, if we have a symplectic manifold , and a point , we can take a (smooth) coordinate chart about . We can then use the coordinate function to push forward to a symplectic form on a neighbourhood of in . If the result holds on , we can compose the coordinate chart with the resulting symplectomorphism to get the theorem in general.
Let . Our goal is then to find a (local) diffeomorphism so that and .
Now, we recall that is a non–degenerate two–form. Thus, on , it is a non–degenerate anti–symmetric bilinear form. By a linear change of basis, it can be put in the standard form. So, we may assume that .
We will now proceed by the “Moser trick”. Our goal is to find a diffeomorphism so that and . We will obtain this diffeomorphism as the time– map of the flow of an ordinary differential equation. We will see this as the result of a deformation of .
Let . Let be the time map of the differential equation
in which is a vector field determined by a condition to be stated later.
We will make the ansatz
Now, we differentiate this :
( denotes the Lie derivative of with respect to the vector field .)
By applying Cartan’s identity and recalling that is closed, we obtain :
Now, is closed, and hence, by Poincaré’s Lemma, locally exact. So, we can write .
We want to require then
Now, we observe that at , so at . Then, as is non–degenerate, will be non–degenerate on an open neighbourhood of . Thus, on this neighbourhood, we may use this to define (uniquely!).
We also observe that . Thus, by choosing a sufficiently small neighbourhood of , the flow of will be defined for time greater than .
All that remains now is to check that this resulting flow has the desired properties. This follows merely by reading our of the ODE, backwards.
|Title||proof of Darboux’s theorem (symplectic geometry)|
|Date of creation||2013-03-22 14:09:55|
|Last modified on||2013-03-22 14:09:55|
|Last modified by||rspuzio (6075)|