In analyzing the spherical pendulum the cotangent space of the sphere is defined as


$ T^\*S^2 = \lbrace (q,p) \in \mathbb{R}^3 \times \mathbb{R}^3; |q| = 1, q \cdot p = 0 \rbrace$ 


my problem with this is that I see the right-hand side of the equation as a set of points, whereas I see the left-hand side as a set of linear functions on the tangent space of $S^2$. 


How can I see them as the same?



Cotangent space of the sphere
What you have written down to me seems to be the tangent bundle, not the cotangent bundle (though they are isomorphic, I suppose).


Consider the projection from your space (let's call it $X$) to the first copy of $\mathbb{R}^3$. This has image exactly those $q \in \mathbb{R}^3$ such that $|q| = 1$ i.e. $S^2$. The fibre over a point $q$ is those vectors $v \in \mathbb{R}^3$ which are perpendicular to $q$; that is, the tangent plane to $S^2$ at $q$.


