Momentum and cotangent space

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arivero

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We are told sometimes that (x,p) is to be considered a point in cotangent space T^*Q. The naivest argument for this is that momentum is a covector, so its transformation law under coordinate changes is just the transformation law of elements in the cotangent space. But for the same token force is a covector, so we also could said that Force lives in T^*Q.

Worse, T^*Q is the dual to TQ, where vector fields live, and in this sense it is compossed of *differential* forms. So it seems that we should see momentum as coordinates of a differential form. But should we?
 

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