Resolving an angular velocity vector into its components

take everything · 15-04-2023 12:25AM #1

Hi,

I'm reviewing rigid body mechanics and angular velocity in particular. Someone might be able to help me with this.

The angular velocity vector:

Now when I take an angular velocity omega in 3 space and decompose it into its x, y and z components, omega_1, omega_2 and omega_3 respectively, we expect the magnitude of the angular velocity vector to be: |omega| = square root of ((omega_1)^2+(omega_2)^2+(omega_3)^2).

However, if you think about the rotation of a body described by the omega vector:

Let's say it does one full rotation in 1 second. Physically, the projections of this rotation onto the yz plane, xz plane and xy plane correspond respectively to the x, y and z components of omega above and each of these projections (of the body onto the respective planes) all undergo one full rotation in 1 second also.

So, on the face of it, the angular velocities of each of these components are on average (over one rotation) the exact same as the magnitude of the angular velocity of our vector omega (if we trace out the motion of the body itself and the motions of the projections of the body):

|Omega| = omega_1 = omega_2 = omega_3

This contradicts the decomposition above.

Can anyone explain what's going on here and what assumptions above are false.

Apart from that, I find it very unintuitive that, although we can resolve an angular velocity vector into components like this, that (at least, averaged over one rotation) the different objects above behave like this.

I've looked for answers online including stack exchange and physics forums (I might be able to link the thread in physicsforums) but in any thread I've seen on the topic, there is either no real consensus (or unnecessary appeals to things like Lie algebra etc are made which offer no physical intuition). It is suggested that angular velocity is a subtle concept not readily amenable to physical intuition, which is worrying to be honest.

The best I've got from reading about this is that it's important to regard the angular velocity and its components instantaneously and not try to place any significance on them all making a full rotation concurrently (I find it hard to just dismiss this physical aspect however).

Anyway, if any physics people can help me understand angular velocity and its components (physically mostly; the mathematics is straightforward vector algebra), I would be quite grateful as I can't get answers anywhere else.

I can also post links to the other thread, or diagrams if clarification is needed.

Thanks.

Just a final note: I know all about how angular velocity is a vector since infinitesimal rotations commute so d(theta)/dt commute. And, interestingly, finite rotations do not commute and are not vectors etc.

Cheers.

Michael Collins · 03-06-2023 06:25PM

When you say a rotation in 3D space, do you mean circular (say) rotation in some fixed plane?

Some diagrams would be nice, if you can.

take everything · 11-06-2023 09:13PM

https://www.boards.ie/discussion/comment/120674448#Comment_120674448

Hi Michael,

Thanks for your reply.

I actually thought nobody would touch this. 😆

Anyway, this is a link to a thread that asks essentially what I'm asking:

https://physics.stackexchange.com/questions/736626/why-dont-the-angular-velocity-components-add-up-for-a-rotation-on-a-tilted-plan

You ask whether it is circular rotation in a fixed plane. To keep things simple, yeah (like in the linked thread). Does it change things if the trajectory was not circular btw.

A related question which I have asked elsewhere, which might obviate asking this one, and which I have found no satisfactory answer for online, is:

Why is angular velocity a vector (I'm not getting into the distinction of vector vs pseudovector incidentally; I just want to understand why angular velocity behaves like a vector).

So the arguments I've seen are just mathematical:

If we can show that infinitesimal rotations are vectors, it follows that angular velocity is a vector. That's fine.

So if you have one infinitesimal rotation, let's call it (I+Adt) and another infinitesimal rotation (I+Bdt) where I is the 3×3 identity matrix and A and B are your 3×3 infinitesimal matrices denoting infinitesimal shift about two given axes, then their product yields I+(A+B)dt. This is easy to show and commutativity is easy to show.

I'm pretty sure this means that these three mathematical objects I+Adt, I+Bdt and I+(A+B)dt then are vectors in a vector space where the binary operation is commutative multiplication since they are isomorphic to matrices A, B and A+B which we know are vectors under the operation of commutative addition.

I was recently reminded of this by a mathematician and this was all fine for me, except for the fact that I don't know if our product above I+(A+B)dt is actually an infinitesimal rotation (physically/geometrically I mean).

I know I+(A+B)dt is a lovely mathematically object that plays nice mathematically with its friends I+Adt, and I+Bdt (both of which I can accept as actual physical infinitesimal rotations) but I see no justification that I+(A+B)dt is an actual infinitesimal rotation physically (geometrically).

Unfortunately I never heard from my mathematician friend again after I asked him would he be able to reconcile the nice maths above with the physics, but I hope a Boardsie can resolve it. 🙂

Honestly though, I've been thinking about this for a while and it has be stumped. I haven't seen an answer to it anywhere (just the simple notion that angular velocity behaves like a vector I'd be happy with tbh).

Texts like Kleppner and Kolenkow cite commutativity of Infinitesimal rotations as sufficient evidence for them to be vectors but that's not true obviously: Commutativity is necessary for an object to be a vector not sufficient.

If someone can put me out of my misery, with either of these queries, I will be eternally grateful.

Thanks

Resolving an angular velocity vector into its components

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