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Comment by colordrops | Hacker News Reader

From a layman's perspevtive I've never understood the existence of angular momentum as anything other than a mental model or abstraction. My intuition tells me that all momentum is only linear (except maybe at the fundamental particle level), and perceived rotation is really just a huge amount of linear interactions by individual particles that make up a larger object. This is similar to how a gas is not really a singular thing but really a bunch of particles bouncing against each other, but can still be described by ideal gas laws as if it were a singular object.

What is the flaw in my thinking?

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Ok so you want the flaw in your reasoning.

Imagine a hula hoop. It’s spinning, but otherwise not moving. Mentally divide it up into lots of little chunks.

Each chunk has a little bit of linear momentum.

1. Is each chunk’s linear momentum constant in time? No, because its direction of motion is changing. Hm, that’s unfortunate.

2. What about if you do R x p, the radius of the loop crossed with the momentum of that little chunk. That gives you a quantity for each chunk that’s unchanging. That’s called the angular momentum of that chunk.

If you add the angular momentum for all the chunks, you get the total angular momentum.

So you see that angular momentum works just like linear momentum. You take a conserved quantity for a single chunk and add up over all the chunks.

You’re free to think of all the little particles and their linear momentum, but that isn’t so useful, since summing the linear momenta gives you zero, whereas summing R x p gives you something to reason about.

From a layman’s perspective, if something spins, you need to apply a force to it to stop spinning, otherwise it will just keep spinning. Hence, conservation of angular momentum. If you change the mass distribution, it will spin faster or slower, just like ice skaters (this is probably the most striking example), but the angular momentum is still conserved, regardless their linear momentum. There is no fundamental difference between this and what happens at the atomic scale.

The not layman’s answer is Noether’s theorem: the invariance of the laws of physics after a rotation of the frame of reference implies conservation of angular momentum.

Which implies that it's separate from linear momentum (the conservation of which is implied by invariance after linear translations).

Indeed!

Your layman’s answer still doesn’t quite do it for me - if you had two masses in space, joined together by a wire, and you span them around their combined centre of mass, angular momentum would remain conserved even if we wound the wire in or payed it out. If I cut the wire, though, wouldn’t the masses immediately shoot off (roughly) in whatever direction they were travelling at the time? That is my interpretation of the “everything seems linear” perspective colordrops offered - it seems like you only get the apparent rotation because of all the other forces holding things together.

edit: I accept that the non-layman’s answer may hold, I just find the intuition a bit off.

Angular momentum would still be conserved when you cut the wire though?

Also, couldn’t you just as well frame all the linear momentum in terms of angular momentum?

... hm, I guess one question is, how would we describe a world where one is conserved but not the other, and visa versa, So e.g. a physics invariant under rotation around each point but not under translations, or visa versa...

Well, it seems like being invariant under rotation around any point, should maybe imply invariance under translations as well..

But, if we are talking about just rotations around a particular point, then a good example is a model of an atom where we consider the nucleus to be the fixed origin, and with the electrons to just be in a rotationally symmetric potential well, and in that model angular momentum is quite important, while linear momentum isn’t quite so important?

It's very decidedly not an abstraction on linear momentum.

It's a conserved quantity just like any other. Light has angular momentum even when travelling in a straight line. Anything with spin does even when stopped.

In physics there is a concept known as symmetry. Any way you can swap around your variables without influencing the answers corresponds to a way in which the laws of physics are symmetric (or a way someone could change the time/position/orientation/etc of the region in which you did your experiment and you would get the same results as long as you stayed in the lab they moved).

Angular momentum is the conserved quantity from rotational symmetry and is just as primary as energy-momentum or charge.

Is that the _same_ angular momentum, or just the same term used for something different? My understanding is that "spin" isn't really anything necessarily spinning and it's more just an analogy for a property of subatomic particles that doesn't exactly correspond to a concept at the macro level.

It has almost the same relationship to angular momentum (and tangentially magnetic fields) as charge on an electron has to charge in a capacitor or an electric fields.

> My understanding is that "spin" isn't really anything necessarily spinning

Yeah, this is kind of the point. A bunch of non-spinning things like particles, or more abstractly an EM field can have angular momentum. This is the same quantity that is conserved in things that are spinning.

> This is the same quantity that is conserved in things that are spinning.

In what sense is it the same? If we called them completely different names, what would be different or more complex?

To flip the spin of a particle from eg. +1/2 to -1/2 it must exchange some angular momentum with its environment, usually in the form of a spin 1 photon. Each unit of spin corresponds to a fundamental amount of angular momentum, and if you add up all of the angular momentum in a closed system the total does not change.

Spin falls out of qm equations as the quantized number associated with the angular momentum quantity that falls out of classical equations.

Much like you get energy-momentum from translating in time and space, and this falls out in a constrained quantum system where energy-momentum is quantized as an integer we call wave number.

IANAP, but it seems to me that right now we are able to make the statement: "Angular momentum is conserved."

If we were to give different names to different types of angular momentum, we could no longer say that. Instead we would have to say something like: "The sum of Type 1 Angular Momentum and Type 2 Angular Momentum is conserved."

Does that qualify as more complex?

Yes to your statement. I'm not quite sure what you're asking with the question, but will try.

I mean you're increasing the complexity of your explanation I guess, and if you don't draw the distinction quite right you might be doing so for no explanatory benefit and a net reduction in communication effectiveness. But there are other reasons quantized spin is important and relevant compared to, say, the net angular momentum you have around Detroit at the current time.

To clarify, my question was basically asking whether the author of the parent comment is satisfied by my comment.

In support of your idea, engineers routinely use FEA simulation software that models dynamic systems without any explicit representation of angular momentum or rotational inertia. Only linear displacement/velocity/acceleration, "translational" mass, and elasticity. Such software can still model rotating systems and all the complex effects that come with rotation. So at least classically, I don't think angular momentum is a fundamentally essential concept.

This isn't explanatory, but should at least demonstrate that it is separate: electrons are point-like particles and thus have no internal substructure to rotate, yet they behave as though they have angular momentum. This "intrinsic angular momentum" is the electron's spin.

That seems like it would violate the conservation of energy. If all the momentum was linear, than changing the direction of those individual particles would require a force.

Changing linear momentum does require a force (but not necessarily work), as does changing angular momentum. In orbits it's gravity, in particle interactions it's the electromagnetic force.

Well, if it completes a full rotation, the applied force over the time of the full rotation is zero (hence, work is zero). So yes, the momentum is linear, but it all pencils out by the direction, right?

That's my understanding of this, but I'm far from a physics expert.

There are forces holding atoms together, such that the conglomerate constitutes a solid object whose geometry is stable in time. Those forces are strong, and are often sufficient to keep a solid object in the same shape even if rotating at high speeds. Sometimes they are not, in which case the solid object is deformed as the individual atoms are "flung outward" by the (ultimately linear) momentum, which overcomes the attractive force that binds atoms into molecules/lattices/whatever.

Thought experiment: if you have an elastic band floating in space, and you nudge it repeatedly to get it rotating faster and faster, will it stretch out (the way a timing belt would if run too fast) or will it stay slack? In its own reference frame, is the band being put under tension — and so experiencing elastic deformation — from the momentum of the particles within it?

Yes it will stretch out.

Lay person’s attempt at an explanation: Centripetal force, but no work is done because the motion of the spinning object is perpendicular to the force continually changing the linear motion to follow the circular course?

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