cotangent bundles as symplectic manifolds

Question

Why do we always think of symplectic manifolds as locally being a contangent bundle, why not simply the tangent bundle, or even easier, why not just $\mathbb{R}^{2n}$?

Answer 1

I can only speak to the context that I see them, so this might not be as robust or deep as you'd like:

A lot of it comes from mechanics. Symplectic manifolds come from a generalization of phase space, which we view mathematically as the cotangent bundle (of configuration space). In fact, the cotangent bundle comes equipped with a canonical one-form $\theta$, and the exterior derivative $\sigma$ of this one-form provides the symplectic structure. The form $\sigma$ allows us to define e.g. Hamiltonian flow, which preserve the volume form under pullback. It is worth noting that Darboux's Theorem says, essentially, that all symplectic forms are equivalent to $\sigma$ on $\mathbb{R}^{2n}$, locally.

In short, relating symplectic manifolds to the cotangent bundle allow us to naturally define a lot of generalizations of and analogues to classical mechanics.