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The angular component of the velocity of a particle in cylindrical coordinates has different units if we consider the vector component $v^{\phi}$ or the one-form component $v_{\phi}$:

$$ v^{\phi} = \frac{x v^y - y v^x}{r^2} ,$$ $$ v_{\phi} = x v^y - y v^x .$$ This came from the fact that we use the Jacobian matrix $J(\rm cart \rightarrow cyl) = \frac{\partial(r,\phi,z)}{\partial(x,y,z)}$ for changing the components of the vectors, while we use its inverse for changing the components of the one-forms.

while $v^{\phi}$ has the dimension of an angular velocity, $v_{\phi}$ has the dimension of a specific angular momentum.

Question: Can we argue something general from this example? Do we have a physical duality between vectors and one-forms, in the sense that one-forms are some conjugate variable of the vectors?

ACA
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  • What do you mean by "physical duality"? There is of course the general duality between vectors and one-forms (musical isomorphism, raising and lowering indices) on (pseudo-)Riemannian manifolds, and it seems you have used exactly that here, using the ordinary Euclidean metric on $\mathbb{R}^3$. That velocities live in the tangent bundles and momenta in the cotangent bundle is also a general notion, see e.g. https://physics.stackexchange.com/q/307794/50583. – ACuriousMind Dec 19 '22 at 11:19
  • I used the therm "physical duality" to distinguish from the musical isomorphism. The vector and one-form are two different physical quantity, in this particular case as an example, why the one-form of a velocity is an angular momentum? I could expect this can happen for many physical quantities encoded in vectors and one form. – ACA Dec 19 '22 at 11:27

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