Now I’ll actually give the proof of the Darboux theorem that a symplectic manifold is locally symplectomorphic to $\mathbb{R}^{2n}$ with the usual form.

Proof of the Darboux theorem

We will prove the equivalent:

Theorem 1 Let ${M}$ be a manifold with closed symplectic forms ${\omega_0, \omega_1}$, and ${p \in M}$ with ${\omega_0(p) = \omega_1(p)}$. Then there are neighborhoods ${U,V}$ of ${p}$ and a diffeomorphism ${f: U \rightarrow V}$ with ${f^*\omega_1 = \omega}$.

The idea is to consider the continuously varying family of 2-forms

$\displaystyle \omega_t = (1-t) \omega_0 + t \omega_1 = \omega_0 + t \alpha$

where

$\displaystyle \alpha = \omega_1 - \omega_0 .$

We will consider a small neighborhood ${U}$ of ${p}$ and a smooth map ${ G: U \times [0,1] \rightarrow M }$ such that ${G_t:=G(\cdot, t)}$ is a diffeomorphism, ${G_0 = id}$, and

$\displaystyle G_t^*(\omega_t) = \omega_0 \ (*).$ (more…)

The purpose of this post is to discuss a few basic facts about differentiable manifolds and state the Darboux theorem, which I will prove next time.  (People who are looking for a more ambitious leap into symplectic geometry might want to try lewallen’s two posts over at Concrete Nonsense.)

A symplectic manifold is a smooth manifold ${M}$ equipped with a closed symplectic 2-form ${\omega}$. In other words, ${\omega}$ is alternating and nondegenerate on each tangent space ${T_p(M)}$

The basic example of a symplectic form is

$\displaystyle \sum_i dx^i \wedge d\xi^i$

on ${\mathbb{R}^{2n}}$ with coordinates ${x^i, \xi^i, 1 \leq i \leq n}$

This can also be written in a more invariant form, which will also give an invariant manner of making the cotangent bundle ${T^*M}$ of any manifold ${M}$ into a symplectic manifold. First, we define a 1-form ${\alpha}$ on ${T^*M}$. Let ${p: T^*M \rightarrow M}$ be the projection downwards. Given ${v \in T^*M}$ lying above ${\xi \in T^*M}$, define

$\displaystyle \alpha(v) = \xi( p_*(v)).$

To make this clearer, here is an interpretation in local coordinates. Let ${x^1, \dots, x^n}$ be local coordinates for ${M}$. Then ${x^1, \dots, x^n, \xi^1, \dots, \xi^n}$ coordinates for ${T^*M}$. Then

$\displaystyle \alpha = \sum \xi^i dx^i$

as is easily checked by working through the definitions. So we can define a canonical 2-form ${\omega}$ as ${\omega = - d \alpha}$; this makes ${T^*M}$ into a symplectic manifold. (more…)