Quote:Original post by arithma
The formula that I posted is the reverse of what I needed [grin]

Quote:
pi(ri) = ri-2 L cross ri

No, that is most definitely p given r under the constraints you required, which is what you asked for. What's the revese? The inverse (r given p)? But r_i(p_i) isn't well-defined. You already have L_i in terms of r_i and p_i. What do you want?

Quote:
I drop any relevancy issues that were raised in face of using orbits. However I praise the avoidance of their usage in the discussion. After all we are trying to figure out an intuitive sense around angular momentum [smile].

It sounds like you already have an intuitive sense. The very fact that you say "Not all p_i = 0" and "Sum({p_i}) = 0" implies "Spinning" shows this.

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Now let's get down to the dirt.

I'm not sure it can be done. Eventually, you're just going to have to pull that (r x p) out of your ass and comment that it doesn't change given the constraints. My recommendation: Try it with just two particles and note that it can be extended to more discrete particles and continuous bodies. Cylindrical coordinates can help.

If you don't want to just pull it out of your ass, you're not going to get there by your route.

OH YEAH
I was once too intrigued by this thing and as far as i can tell:
angular momentum is NOT derived from linear momentum.

but i'll give you a clue.
imagine a long rigid body and a force that acts on a tip of it making it spin.
how does energy/linear momentum propagates between points/particles from one tip to the other? no idea? me neither. and that would be i belive the answer to your question.

Quote:Original post by arithma

It actually derives angular momentum from linear momentum and a cross product.. [grin]

Ahem
It does not. It derives angular momentum of a rigid body (system of points)
from angular momentum of a point.

Quote:Original post by arithma
I have actually found a very interesting link that backs me up. It talks about Rigid Body & Linear Momentum & Angular Momentum & Torque.

It actually derives angular momentum from linear momentum and a cross product.. [grin]

He doesn't derive it, he defines it. After getting to equation (3) he says:

Quote:
Define:

L(t) = Sum_{i = 1}^n (m_i r_i(t) x r`_i(t))
T(t) = Sum_{i = 1}^n (r_i(t) x F_i(t))

And to even get to this point he says:

Quote:
Now take the cross product of r_i(t) and equation (1_i) and sum over i.

Like I said, you have to pull it out of your ass.

I know the trouble with finding the explanation that works for you. I've often had talk to a couple professors (or fellow students) before getting an explanation that worked for me. I honestly didn't think the sum over i was the part you were struggling with. I thought you were having trouble with the conservation of angular momentum and the fact that the cross product will never just pop out of manipulations on the linear momenta. Glad you finally found the explanation that works for you.