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Emailanmol's discussion on angular displacement 
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#19
Mar2712, 10:26 PM

Sci Advisor
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.



#20
Mar2712, 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. 


#21
Mar2812, 08:26 AM

Mentor
P: 15,065

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 calculusbased freshman physics text. It is not a high school level text. Rotation Rotations in ℝ^{n} are described by the n×n 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 ℝ^{2}, and ℝ^{2} only. Go beyond ℝ^{2} 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 ℝ^{2}  and rotations in ℝ^{2} are commutative.) Rotations in ℝ^{3} 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 ℝ^{3}. It doesn't work in ℝ^{2}; rotations in ℝ^{2} are about a point, not an axis. It doesn't work in ℝ^{4} or higher, either. Bottom line: This concept of an axis of rotation is unique to ℝ^{3}. Since rotations in ℝ^{3} can be described in terms of an angle (a scalar) and a rotation axis (a direction), does this mean that rotations in ℝ^{3} 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 ℝ^{3} are not commutative, so even though rotations in ℝ^{3} can be represented by something that superficially looks like a vector in ℝ^{3}, 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:
Angular velocity First we need to define what one means by "angular velocity". Since infinitesimal rotations in ℝ^{n} can be treated as vectors in ℝ^{n*(n1)/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 ℝ^{n} is that they are skew symmetric matrices in ℝ^{n}. A skew symmetric matrix in ℝ^{n} has n*(n1)/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 ℝ^{n} 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 looseygoosey, hand waving physics math is needed. Unfortunately, the mathematics is a bit hairy. Back to ℝ^{3}: A skew symmetric matrix in ℝ^{3} 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 ℝ^{3} 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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