Can an angle be considered to be a vector?

[lugita15]

First off, your analysis in post #27 is incorrect. Differential angle does not commute to all orders.

Yes, it is certainly true that infinitesimal rotations commute to first order but not to second order. But I would like to find the error in my proof that they DO commute to all orders.

The fundamental reason rotations don't commute is because they aren't vectors. They are instead rotation matrices.

But the reason why we represent rotations as noncommuting matrices is because we know in advance that they do not commute. Surely we would represent them in an abelian group if we happened to believe that they do commute. So this does not strike me as a fundamental reason.

To me, it seems like the reason rotations form a nonabelian Lie group is because they are exponentials of an anti-commuting Lie algebra. So perhaps the real question is why do we make the Lie bracket anti-symmetric? Or to put it another way, why are the cross product of vectors and the wedge product of differential forms anti-commuting? (And no, I wouldn't consider the fact that the cross product can be represented in terms of a skew-symmetric matrix to be a good enough explanation, because why don't we define a product of vectors that is not represented by a skew-symmetric matrix?)

You've done the experiment with the book. Believe your eyes.

The book demonstration is sufficient to convince me THAT rotations don't commute, but it does not suffice as an explanation for WHY I'm seeing what I'm seeing.

 

[D H]

Yes, it is certainly true that infinitesimal rotations commute to first order but not to second order. But I would like to find the error in my proof that they DO commute to all orders.

You didn't state your proof mathematically. Try doing so. It won't work.

But the reason why we represent rotations as noncommuting matrices is because we know in advance that they do not commute.

Exactly. That's physics for ya. A physicist's job is to describe reality.

To me, it seems like the reason rotations form a nonabelian Lie group is because they are exponentials of an anti-commuting Lie algebra. So perhaps the real question is why do we make the Lie bracket anti-symmetric?

To me it seems that the best place to start is linear algebra. Euler did quite a bit studying rotation without the benefit of linear algebra. Linear algebra just makes Euler's work easier. The connection between the transformation matrices in linear algebra and Lie groups is an after-the-fact development and in a sense makes things harder. You won't see much on Lie algebras and Lie groups in physics until you get to grad school. The mathematics is not easy. The mathematics of linear algebra can be taught to some extent at the high school level.

The time derivative of a time-varying transformation matrix can be represented as the matrix product of the transformation matrix and a skew symmetric matrix or as the matrix product of a different skew symmetric matrix and the transformation matrix. A 3x3 skew symmetric matrix can be represented by an axial vector, and this is the reason why we can represent angular velocity in three space as a vector. Note well: This only works in ℝ³ only. For example, a 2x2 skew symmetric matrix has one free parameter, so angular velocity in ℝ² can be represented by a scalar (a pseudoscalar to be precise) rather than a two vector. A 4x4 skew symmetric matrix has six free parameters, so angular velocity in ℝ⁴ cannot be represented by a four vector.

Or to put it another way, why are the cross product of vectors and the wedge product of differential forms anti-commuting?

We use the cross product of vectors in ℝ³ (the vector cross product as-is is unique to ℝ³) because it is a useful concept. If it wasn't useful we wouldn't use it. Physicists use mathematical tools that help in the endeavor of describing reality.