# If momentum is a covector, how does $p=mv$?

There are several explanations on this site [1] [2] [3] about why momentum is a covector while velocity is a vector. This distinction is important for the geometric description of classical mechanics.

However, none of these explanations reconciles this with the seeming contradiction that $$p=mv$$, which on its face suggests that momentum and velocity are the same type of object. How can we resolve this apparent contradiction?

I would expect that $$p=mv$$ exploits an isomorphism between the tangent and cotangent spaces, allowing us to represent $$p$$ as a vector even though it is "naturally" a covector. But if that's the case, how is this isomorphism defined, and where does it enter into the formalism of classical mechanics?

2. Example: If $$L=\frac{1}{2}m_{ij}v^iv^j-V$$, then $$p_i=\frac{\partial L}{\partial v^i}= m_{ij}v^j$$. Often the mass metric structure is of the form $$m_{ij}=m~g_{ij}$$.
• $\uparrow$ In many cases. Commented Aug 20, 2022 at 4:50