Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by definition of the tangent bundle, we have $(x,x') \in TM$.
Now, in classical mechanics we learn that the conjugated mometum $p(x,x') = \partial_2L(x,x')$ and now I read that this guy is an element of the cotangent space, but I have no idea why.
I mean, to be in the cotangent space, you need to take elements in the tangent space of $x$ which are velocities as arguments and only depend linearly on them. Although it is clear that p takes velocities as arguments which is alright, it is not clear to me at this moment why this should happen linearly. Is this an additional (physical) input at this point or is there a mathematical argument why the momentum is now a linear map?
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