Unit Norm Axis Rotation in R3: Exploring Representation & Algebra

In summary, the conversation discusses the representation of rotations in R3 using matrices and euler angles. The author talks about the canonical basis and its relation to the representation A, as well as the construction of a unitary matrix using the operators A_i. The conversation ends with a discussion on the difficulty of evaluating the exponential in general, but notes that for the 2d representation of SO(3), there are easier expressions for the matrices.
  • #1
LCSphysicist
646
162
TL;DR Summary
I am having trouble to understand the reasoning that the author follows in these pages.
A rotation about an unit norm axis in R3
1624685734661.png
can be given by the following, that gives basically the elements of the matrix of the rotation:
1624685272932.png

With these expression, we can obtain the three infinesimal rotations that corresponds to the three euler angles by first writing the three individuals matrix of these rotations, and then, taking the derivative:
1624685381104.png

Now, the autor gives the algebra regardind these transformations and talk about its representation in Rn
1624685431887.png

It is said that the representation A have the same algebra that g. And so he starts to talk about the canonical basis to represent as a way to write the representations.

1624685476549.png

After that, he returns to the beginning and says that
1624685560659.png

The relatios six and seven he says is that:

1624685593061.png

Now, i am extremelly confused about all this thing. More preciselly, i can't understand how 1.29 was obtained. It was used the A representation? How do he uses it? There is something to do with the canonical basis?
 
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  • #2
Since you want to obtain a 2d representation, you need to use the representation with ##2=2j+1## i.e. ##j=1/2##. Then the "canonical basis" is 2d and consists of ##f_{-1/2}, f_{1/2}##. With eq. 1-28 you now know how ##L_3## and ##L_{\pm}## act on that base, and therefore you can construct the matrices in this base.
Now, knowing these 3 operators you can easily compute, using 1-26 the operators ##A_i##. And from that, you can construct the unitary matrix.
 
  • #3
Gaussian97 said:
Since you want to obtain a 2d representation, you need to use the representation with ##2=2j+1## i.e. ##j=1/2##. Then the "canonical basis" is 2d and consists of ##f_{-1/2}, f_{1/2}##. With eq. 1-28 you now know how ##L_3## and ##L_{\pm}## act on that base, and therefore you can construct the matrices in this base.
Now, knowing these 3 operators you can easily compute, using 1-26 the operators ##A_i##. And from that, you can construct the unitary matrix.
Hello. Thank you. The real problem to me is, following your reasoning, make the last step. Tthat is, from the Ai obtained, how to construct the unitary matrix. How do exact we could do it? I am still new in these representations things, so i am a little confused about that.
 
  • #5
Given the set of generators ##A_i##, you can construct the elements of the group as
$$g(\alpha, \beta, \gamma) = \exp{(\alpha A_1 + \beta A_2 + \gamma A_3)}$$
Where the exponential must be understood as its Taylor expansion. The evaluation can be, in general, very difficult, fortunately for the 2d representation of SO(3) you can find easy expressions for the matrices ##(\alpha A_1 + \beta A_2 + \gamma A_3)^n## which allows you to write the exponential in the form 1-29
 

Related to Unit Norm Axis Rotation in R3: Exploring Representation & Algebra

1. What is Unit Norm Axis Rotation in R3?

Unit Norm Axis Rotation in R3 is a mathematical concept that involves rotating a three-dimensional object around a fixed axis while maintaining its unit norm. This means that the length or magnitude of the object remains constant throughout the rotation.

2. How is Unit Norm Axis Rotation represented?

Unit Norm Axis Rotation is commonly represented using a rotation matrix, which is a square matrix that describes the transformation of coordinates in a three-dimensional space. It can also be represented using quaternions, which are four-dimensional numbers that can be used to describe 3D rotations.

3. What is the purpose of Unit Norm Axis Rotation?

Unit Norm Axis Rotation is used in various fields such as computer graphics, robotics, and physics to describe and manipulate three-dimensional objects. It allows for precise and efficient rotations of objects while maintaining their shape and orientation.

4. How is Unit Norm Axis Rotation calculated?

The calculation of Unit Norm Axis Rotation involves using mathematical formulas and algorithms to determine the rotation matrix or quaternion that describes the rotation. These calculations take into account the angle of rotation, the axis of rotation, and the initial position of the object.

5. Are there any real-world applications of Unit Norm Axis Rotation?

Yes, Unit Norm Axis Rotation has many practical applications in fields such as computer graphics, video game development, animation, and robotics. It is used to create realistic and smooth animations, control the movement of robots, and simulate physical phenomena in virtual environments.

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