Quote:Original post by Way Walker
Look at it this way, if the body is at rest but rotating, then the sum of the linear momenta will be 0 by definition, so finding a formula for the linear momentum of each particle is not equivalent to finding the angular momentum.

Well am not proposing to find a relation between "sum of linear momentum" and "angular momentum". What am proposing is starting the reasoning from the (1) body being rigid and (2) the sum of linear momentum is zero. If these conditions must be satisfied, then the body MUST be rotating. Relying on this we MUST be able to find a formula for p given r (using same meanings for symbols). I believe if somebody may be able to find such a relation, it will provide instantaneous insight into angular momentum...

Moreover, you must understand that angular momentum is definetly not a concept by itself and apart from linear momentum. It is only a simplification. That emerges from the fact the Newton's Three Laws are enough to explain any phenomenon in classical mechanics; thus angular momentum is a derived concept. Yet we never see any literature concerning the "DERIVATION" of a concept. This was expectable at high school but I never imagined that it may prevail in the engineering physics courses. I don't know about physics majors: so is this really the situation?

Quote:Original post by arithma

Well am not proposing to find a relation between "sum of linear momentum" and "angular momentum". What am proposing is starting the reasoning from the (1) body being rigid and (2) the sum of linear momentum is zero. If these conditions must be satisfied, then the body MUST be rotating. Relying on this we MUST be able to find a formula for p given r (using same meanings for symbols). I believe if somebody may be able to find such a relation, it will provide instantaneous insight into angular momentum...

Moreover, you must understand that angular momentum is definetly not a concept by itself and apart from linear momentum. It is only a simplification. That emerges from the fact the Newton's Three Laws are enough to explain any phenomenon in classical mechanics; thus angular momentum is a derived concept. Yet we never see any literature concerning the "DERIVATION" of a concept. This was expectable at high school but I never imagined that it may prevail in the engineering physics courses. I don't know about physics majors: so is this really the situation?

Perhaps this is a case of putting the horse behind the cart... the conservation laws are more general than Newton's laws and are given higher precedence (here). In other words, Newton's laws should be derived from conservation of momentum, not the other way around.

[edit] perhaps Noether's theorem is what you are looking for?

Quote:Original post by Way Walker
Consider an object in a force field (e.g. gravity, part of a rigid object, etc.). Let's say it's in orbit. It's orbiting so obviously momentum isn't conserved. However, it won't just spontaneously stop spinning, so something's being conserved. Energy? Maybe, but if its orbit contracts (think figure skater) that won't be conserved. The quantity (r x p) is conserved. This is somewhat intuitive if you picture an orbitting object; in an elliptical orbit v (and thus p) decreases with r. You don't escape this problem with a circular orbit because the direction of v changes, but (r x p) will always be normal to the orbitting plane. Basically, r "corrects for" the changes in p.

Quote:Original post by arithma
I also drop my questions about correlating the conservation of angular momentum to conservation of energy... It is obvious that it an invalid question when compared to linear momentum and Linear Kinetic Energy which are not necessarily conserved together.

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Quote:Original post by Way Walker
Given L and r, you can't find p. Cross product isn't reversible....

If you follow me precisely as ask the question, I'm concerned with the transition from linear momentum to angular momentum. So your note over here may be precise but a little bit irrelevant.
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And regarding the orbits:
Orbits are also irrelevant since am considering total linear momentum is zero; except if you're talking about a wider system that spans other objects, but again it loses the "Rigid Body" qualification...

Quote:Original post by Way Walker
This is going to sound critical, but it's not, I'm honestly curious: Did you demand a rigorous definition of limits before you were satisfied with derivatives?

Not really, because we learned about it in eleventh grade, so I didn't have enough backward curiosity if I may say. And besides, for most applications, an empirical understanding of small entity is achievable

Quote:Original post by Way Walker
By the way, you didn't play my game: describe momentum to me.

I assume you mean linear momentum. If that's it then it is easy. An understanding of mass as "mass of particles" then as "quantity of mass of particle" and directly map "mass of particle" into "particle" so you have it as "quantity of particles". And velocity doesn't need to be explained, (the differential of position over the differential of time...). Now if you assume each particle has an equal linear momentum to each other particle (no rotation) then linear momentum can be understood as: the "quantity of particles" * "velocity" which could be manipulated to "quantity of velocity of particles". Viola; At least it makes sense to me :)

Quote:Original post by jjd
Perhaps this is a case of putting the horse behind the cart...

According to your source, the validity is given the higher precedence, not actually the concept itself, which is related to conditions such as approaching speed of light and stuff; we are discussing a Classical Mechanics Universe

Quote:Original post by arithma

According to your source, the validity is given the higher precedence, not actually the concept itself, which is related to conditions such as approaching speed of light and stuff; we are discussing a Classical Mechanics Universe

Um, what? Read the seciton entitle "Relationship to the conservation laws", it is about classical physics and has nothing to do with einstein's relativity. Also, you are trying to squeeze the conservation of angular momentum out of newton's law, which are derived from conservation of linear momentum... and you wonder why you're having trouble understanding it?!

I give up.

Angular momentum as we define it is a derivation. So why do we care about it, as it's arguably not as fundamental as other quantities ?

Simply because in rotational problems it's the quantitiy most often conserved. E.g. a freely rotating non-symmetric rigid body will have an axis of rotation, a speed of rotation, and so an angular velocity. In general none of these is conserved - all can change over time, and in general do for irregular body. But angular momentum IS conserved, providing no external torque is applied.

As it can be related to other quantities it can tell us things about them which are difficult to deduce in other ways. In some ways the fact that it's conserved makes it more fundamental. E.g. fundamental particles have a fixed 'intrinsic angular momentum', also called spin.

You might also consider kinetic energy, but in general is not conserved in non-rigid bodies. E.g. a planet orbiting a star gains and loses kinetic energy as it moves closer to and away from the star. Or in a non-elastic collision between two bodies energy is lost but providing there are no external forces or toruqes total linear and angular momentum are conserved.

To jjd:

Quote:From the given wiki source
The laws of conservation of momentum, energy, and angular momentum are of more general validity than Newton's laws, since they apply to both light and matter, and to both classical and non-classical physics. In the special case of a system of material particles interacting via instantaneously transmitted forces, Newton's second law can be viewed as a definition of force, and the third law can be derived from conservation of momentum.

And if you were right and I was wrong (in any sense you can put it) then I am sorry for any discomfort I did induce.. Thanks for your time anyway [grin]

To johnb:
I am not discussing "the usefulness of angular momentum" but asking about "the origins of its usefulness" by investigating its mathematical origin.

Quote:Original post by Way Walker
The only other interpretation I can think of is for a formula for p given a position r within the body, but surely that depends on the system you're considering.

This is what I meant.

Quote:Original post by Way Walker
its "rigidness" implies a force pulling each portion of the body toward the center of rotation (otherwise it would fly apart).

Ok. Then I correct my statement to "non-circular orbits" are irrelevant. I think this is fair enough.

Quote:Original post by Way Walker
Given L and r, you can't find p. Cross product isn't reversible.

For rigid bodies it is reversible.
vi = (m ri2)-1L cross ri

The formula that I posted is the reverse of what I needed [grin]

In addition, I am not asking about the validity of the formula of angular momentum. I am asking about the steps that get you from "linear momentum of particles" into "angular momentum of rigid body". This is assuming you don't use nothing but math and the concept of "linear momentum".

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].

Now let's get down to the dirt.
For a translationally resting rigid body the following is observed:

- Sum of pi = 0; which we get from the "translationally resting" constraint, which is only for the sake of simplification
- Every particle is connected to every other particle by a nonelastic rigid rod; to satisfy the rigidity constraint

Manipulation steps: x, y, z,...
to reach a situation where miraceously a quantity appears that simplifies things: ANGULAR MOMENTUM. This quantity must appear somewhere since the latter conditions allow rotational movement exclusively.

The outcome will allow us, instead of having to use the function pi(ri) to describe the rotation of the body, to use the constant quantity L that arrises from ??? to describe that rotation. Then the quantities pi are easily calculated given their ri and the constant alleged quantity.

There are two other things: (1) that "using system of equations of linear momentum of particles is equivalent to using the angular momentum of rigid body" to describe the rotation and (2) an answer to a similar (but simpler) question concerning the linear momentum of a rigid body.

(1)
We can describe the rotation of a rigid body (implicitly) by the following funtion:
pi(ri) = ri-2 L cross ri
This is of course in a confined situation where r and L and p are orthogonal in-pairs; but you get the idea.
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(2)
How is the concept of linear momentum of a rigid body established:
Constraints: resting rotationally

resting rotationally: vi = v for any i in the body. Interestingly, that preserves the rigidness of the body.

{
sum of pi = sum of ( mi vi )
ptotal = sum of ( mi v )
ptotal = sum of ( mi ) v
ptotal = mtotal v
}
These steps are analogues to {x,y,z,...} in the answer that I seek

Please note that I allowed my self to use linear momentum of particles here because it is of low intuitive requirements. However, you shouldn't allow yourself to use angular momentum of particles in the establishment of angular momentum of a rigid body. This is because it is assumed that the whole notion of "angular momentum of anything" is not known before this step.

Please bare with me...