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Emailanmol's discussion on angular displacement

by emailanmol
Tags: angular, discussion, displacement, emailanmol
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Mar27-12, 10:26 PM
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P: 2,470
Yes, for small angles it is true in 3D, but that's not the same thing as saying that instantaneous angular displacements are additive. You can say that instantaneous angular velocity is integrable, of course, which is very useful, but it doesn't make displacements themselves vectors.

I think that this problem is arising from the fact that perhaps the book is for high school, and clearly the terms we are dealing with are explained and defined in much more detail in the higher branches of physics.

Is this perhaps why the book is going wrong?
Could be. The vector space is defined sufficiently well here. You can easily test if each of the axioms applies to your quantity to see if it's a vector or not. It's a little hard to say if these are the exact criteria author has had in mind.
Mar27-12, 10:31 PM
P: 297
NOTE : I had edited my previous post a bit, but it changes the meaning in no way :-)
Ok, thanks K2 . I will go through the wiki page.
Mar28-12, 08:26 AM
P: 15,202
There's quite a few misunderstandings in this thread, which is one of the reasons I separated it from the homework help thread where this thread originated.

First off, Halliday, Resnick, and Krane Physics Volumes I and II is a calculus-based freshman physics text. It is not a high school level text.

Rotations in ℝn are described by the nn proper (or special) orthogonal matrices. These special orthogonal matrices form the group SO(n). Rotations are not vectors. They are matrices. Rotation matrices are commutative in ℝ2, and ℝ2 only. Go beyond ℝ2 and rotation A followed by rotation B is generally different from rotation B followed by rotation A. (They are the same if rotations A and B are parallel to the same plane. In this special case, the rotations are just a pair of rotations in ℝ2 -- and rotations in ℝ2 are commutative.)

Rotations in ℝ3 can be described in terms of aa rotation by some angle [itex]\theta[/itex] about some axis of rotation [itex]\hat u[/itex]. This only works in ℝ3. It doesn't work in ℝ2; rotations in ℝ2 are about a point, not an axis. It doesn't work in ℝ4 or higher, either. Bottom line: This concept of an axis of rotation is unique to ℝ3.

Since rotations in ℝ3 can be described in terms of an angle (a scalar) and a rotation axis (a direction), does this mean that rotations in ℝ3 are vectors? The answer is no. Having a magnitude and a direction are necessary but not sufficient conditions for some thing to qualify as a "vector." Vectors have to add per vector addition rules. In particular, vector addition is commutative. Rotations in ℝ3 are not commutative, so even though rotations in ℝ3 can be represented by something that superficially looks like a vector in ℝ3, this angle+axis representation is not a vector.

So what is this stuff in Halliday, Resnick, & Krane about small rotations being vectors? That is not what the book says. The book is very clear: "finite angular displacements cannot be represented as vector quantities." The book does some handwaving to arrive at the result "infinitesimal angular rotations can be represented as vectors." Some things to note here:
  • Infinitesimal ≠ small. Small is still finite. Small angular displacements cannot be represented as vector quantities.
  • There's a lot of handwaving in getting to that result. This is understandable; the underlying math needed to make this rigorous is far, far beyond freshman level math/physics.
  • This trick only works in ℝ3. In ℝ2, infinitesimal rotations can be represented by a single parameter. Six parameters are needed in ℝ4, ten in ℝ5, and in general n*(n-1)/2 parameters are needed to describe an infinitesimal rotation in ℝn.
  • The n*(n-1)/2 parameters needed to describe an infinitesimal rotation in ℝn act a lot like a vector in ℝn*(n-1)/2. They add like vectors, they scale like vectors. In this sense, they are vectors.

Angular velocity
First we need to define what one means by "angular velocity". Since infinitesimal rotations in ℝn can be treated as vectors in ℝn*(n-1)/2, one obvious definition of angular velocity is to use this concept of infinitesimal rotations. (Note well: This is handwaving physics math, the kind of stuff that makes mathematicians cringe.)

An even better way to look at these infinitesimal rotations in ℝn is that they are skew symmetric matrices in ℝn. A skew symmetric matrix in ℝn has n*(n-1)/2 independent values -- the exact number of independent values in our infinitesimal rotation. Viewing angular velocity as a skew symmetric matrix is IMO the best way to envision angular velocity in general. This representation is deeply connected to the Lie algebra that generates SO(n).

This leads to an even better way to envision angular velocity. Forget the infinitesimal rotation stuff. The time derivative of a time varying rotation matrix in ℝn can be represented as the matrix product of that rotation matrix and some skew symmetric matrix, or as the matrix product of a different skew symmetric matrix and the rotation matrix. This is very generic. No loosey-goosey, hand waving physics math is needed. Unfortunately, the mathematics is a bit hairy.

Back to ℝ3: A skew symmetric matrix in ℝ3 has three independent parameters which can be represented as a three vector if the resulting three element construct truly does act like a vector. It does. Angular velocities in ℝ3 have a magnitude and direction, they add per the rules of vector addition and they scale per the rules of vector multiplication by a scalar. That is all that is needed to say that these things are "vectors." There is one way that angular velocities differ from things such as position and velocity vectors, and that is how they behave upon reflection. In this sense, angular velocities and angular accelerations are better described by as being pseudovectors or axial vectors.

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