Return to Revisions

Well, I'll not say anything deeply new here - just a (hopefully correct) summary.
Suppose you are given your favorite manifold, say $Q.$ Then its cotangent bundle $M=T^* Q$ comes equipped with some canonical structure (Keep in mind that the cotangent bundle $T^* Q$ and the tangent bundle $TQ$ are isomorphic as vector bundles over $Q$ (in particular their total spaces are diffeomorphic, but not canonically so), but for some miraculous reason the cotangent bundle has "more" structure). Denote by $\pi:T^* Q\rightarrow Q$ the projection, associating to a covector its basepoint. Differentiating this yields the tangent map of $\pi$, $T\pi: T(T^* Q)\rightarrow TQ$. With its help one can define a one-form $\theta$ on $T^* Q$ (usually called "canonical" or Liouville one-form). It is defined via
$\theta_\alpha:T_\alpha (T^* Q)\rightarrow \mathbb{R},$ $v\mapsto \alpha(d\pi(\alpha).v)$
for any point $\alpha\in T^* Q$. To explain why the definition makes sense: $v$ is an arbitrary element of $T_\alpha (T^* Q)$, the differential of $\pi$ at $\alpha,$ $d\pi(\alpha)$ is a linear map from $T_\alpha (T^* Q)$ to $T_{\pi(\alpha)} Q$. Furthermore $\alpha$ can be interpreted as a linear form on the tangent space of its base point; consequently it makes sense to evaluate it on $d\pi(\alpha).v$.
Now the symplectic form $\omega\in \Omega^2(T^*Q)$ is defined as the exterior differential of $\theta$ (some authors prefer to smuggle a minus sign in). Notice $\omega$ is defined purely intrinsically (no choice of coordinates for instance, even though one often sees expressions like $\theta=p_idq^i$).
That's the reason why you can associate a symplectic manifold (aka phase space) to any manifold (aka configuration space). But to elaborate on what José Figueroa-O'Farrill already said: there are symplectic manifolds which are not of the form $T^* Q$ for some $Q$. Probably the easiest example are closed symplectic manifolds, i.e. compact ones without boundary (they occur for instance as surfaces with a fixed volume form or as nonsingular complex projective varieties together with the Fubini-Study form). You can easily show that their symplectic form can not be exact (that is, of the form $d\theta$ for some one-form $\theta$) unless they are zero-dimensional. Because if it were exact, by Stokes' theorem the integral $\int \omega^{\wedge (dim M/2)}$ would have to be zero. And it is difficult to find manifolds with zero total volume!