What is the Relationship Between Momentum and Cotangent Space?

[arivero]

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?

 

[Evalesco]

I think it is important to understand the distinction between momentum and force. Momentum is a conserved quantity, meaning that it is invariant under coordinate transformations. On the other hand, force is not a conserved quantity and its transformation law under coordinate changes is different. So in this sense, momentum can be considered to live in T^*Q while force cannot.