Physical interpretation of one of Hamilton's equations

[Syrus]

Homework Statement

I've attached a picture from a passage of my book (Liboff, Quantum Mechanics) with which I am having difficulty. Specifically, equation 1.25 claims to possesses a certripetal force factor (in the text underneith) and a moment arm factor. I see both of these terms present. However, shouldn't the centripetal force factor reduce to mv²/r? Also, i assume the moment arm is included because torque is formally (clasically) r x F. However, here θ denotes the polar angle, and so rcos(θ) is the z-component of r. As a result, I don't see how this expression amounts to the torque. Can someone clarify this for me?

Homework Equations

The Attempt at a Solution

 

[Syrus]

Okay, I may have made some progress already:

For everything below, phi = σ.

In this case, mv²/r is equal to m(rσ)²/r = mrσ². Now, r x F = |r| |F| sin(ζ), where ζ is the angle between r and F. But this must mean here that sin(θ)cos(θ) = sin(ζ), and this is (perhaps?) what I don't see.

**Upon second thought, this centripetal force arises from motion in the phi direction at constant r and theta, so the cross product is simply the product of the magnitudes r and F (since the angle between these two vectors is always 90 degrees). Thus, disregard my comments above.

 

[Syrus]

Still looking for help!