Vectors, one-forms and gradients

[nrqed]

well if you have to use a metric to view momentum as a covector, that suggests it is really a vector. as it seems it should be if it is proportional to the velocity vector.

?

I don't understand. If there is a metric, there is a natural correspondence between vectors and covectors, no? And if we use the metric to obtain a covector from a vector, the covector is a bona fide covector, no?

And since the metric is not in general a constant, the generalized momenta will not be proportional to the generalized velocities in general (the generalized momenta do not even have the for of $m {\vec v}$ in general, where here v is the velocity vector in the usual high school sense.

By the way, the genralized velocities and momenta are not necessarily the velocities and momenta we learned in elementary physics. They don't even have the correct dimensions, in general.

You see how confusing things are for a physicist trying to learn the stuff?
It's more difficult than learning things from scratch especially given that books rarely show explicitly the correspondence between vectors, forms, metric and everything we have learned before. That may sound like "symbol pushing" ot you but some of that is required in order to really understand things, imho. (adding to the the whole business of integrating over forms does not help)

Going back to an example. For a particle moving in two dimensions in a central potential (let's say),

$$L = {m \over 2} ({\dot r}^2 + r^2 {\dot \theta}^2) - V(r)$$
The generalized velocities have components ${\dot r}$ and ${\dot \theta}$. So the metric is diagonal(m, m r^2).

The generalized momenta (covectors) have components $p_r = m {\dot r}$ and $p_\theta =m r^2 {\dot \theta}$.

with those definitions, "feeding" the generalized vector field ${\dot q}_r \partial_r + {\dot q}_\theta \partial_\theta$ to the covector $p_r dr + p_\theta d\theta$ gives half the kinetic energy.

 

[mathwonk]

i guess if you can argue that the metric is somehow natural then a vector that uses the metric t be viewed as a covector is somehow naturllly a covector.

but if you have ametric there is essentillyno difference between a vector and a covector. but metrics are not so natural.