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…)