Cross Product in Spherical Coordinates - Getting conflicting oppinions

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Discussion Overview

The discussion centers around the application of the cross product in spherical coordinates, particularly in the context of calculating the angular momentum of a particle. Participants explore whether it is feasible to perform these calculations directly in spherical coordinates or if conversion to Cartesian coordinates is necessary.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants note that Wolfram and various sources claim there is no straightforward method to perform the cross product in spherical coordinates, suggesting conversion to Cartesian coordinates is required.
  • Others, including some professors and textbooks, argue that it is possible to perform the cross product directly in spherical coordinates, as long as one considers the instantaneous directions, which are mutually perpendicular.
  • A participant mentions that their calculations in spherical coordinates yield results consistent with those obtained in Cartesian coordinates, indicating that the method may be valid under certain conditions.
  • There is a discussion about the correct order of the coordinate directions (r, theta, phi) when forming the determinant for the cross product.
  • One participant provides a formula for angular momentum in spherical coordinates, suggesting that it can be derived without converting to Cartesian coordinates.

Areas of Agreement / Disagreement

Participants express conflicting opinions regarding the validity of performing the cross product in spherical coordinates. No consensus is reached, as some support the direct approach while others advocate for conversion to Cartesian coordinates.

Contextual Notes

Participants highlight the need for clarity on the conditions under which the cross product can be performed in spherical coordinates, as well as the implications of using different coordinate systems for calculations.

Damascus Road
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Hey all,
I really need some clarification here.

I've seen problems dealing with the Angular Momentum of a particle, working in spherical coordinates. Wolfram says that there is no simple way to perform this and do the determinant, and you will find many people and other websites claiming this. i.e. you must convert to Cartesian, or I've also seen some kind of operator.

However,
Some of my own professors have said you MAY do this. There are also E-mag textbooks, etc. that do the cross product in spherical coords!

In my own meddling, it seems like it works fine, as long as you look at an instant in time, since the directions are not constant, but they are however, all mutually perpendicular to each other. When I do this, it is identical to what I would get in Cartesian.

Also, if this is allowed, can someone tell me in what order the directions should be in the determinant? (r, theta, phi)
(phi being in the x,y plane)
 
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Maybe W and others were referring to Del X something. The cyclic order r, theta, phi is correct, with anyone of them going first.
 
Damascus Road said:
Hey all,
I really need some clarification here.

I've seen problems dealing with the Angular Momentum of a particle, working in spherical coordinates. Wolfram says that there is no simple way to perform this and do the determinant, and you will find many people and other websites claiming this. i.e. you must convert to Cartesian, or I've also seen some kind of operator.
...

Hmm ... I did a back-of-the-envelope caliculation and got myself

L = m r 2 * (d\theta/dt * phi-direction - sin \theta * d \phi/dt * theta-direction )

Of course, you could convert it into Cartesian, but it's easiest to do it in the coordinate system in which your trajectory is given.
 
What you wrote, xlines, is your result from the cross-product?
 
Damascus Road said:
What you wrote, xlines, is your result from the cross-product?

Yes, that is \vec{L} = m \vec{r} x \vec{v} in spherical coordinates, angular momentum of a pointlike particle.
 

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