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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?

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1 Answer 1

up vote 2 down vote accepted

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 $q$.

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Furthermore, the usual embedding $\: S^2 \to \mathbb{R}^3 \:$ and the standard inner product on $\mathbb{R}^3$ combine to $\hspace{.6 in}$ induce an isomorphism from the tangent bundle to the cotangent bundle. $\;\;$ –  Ricky Demer Nov 12 '12 at 22:41
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