Two successive rotation (Goldstein problem 4.13)

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In summary, two successive coordinate rotations through angles ##\Phi_1## and ##\Phi_2## can be considered as the sides of a spherical triangle with the angle opposite to ##\Phi## given by the angle between the two axes of rotation. This can be seen by constructing a diagram of a spherical apple, where the angle between the rotation axes is equivalent to the angle between the planes of rotation and the angle opposite to ##\Phi## is the angle between the planes of rotation.
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arpon
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Homework Statement


Suppose two successive coordinate rotations through angles ##\Phi_1## and ##\Phi_2## are carried out, equivalent to a single rotation through an angle ##\Phi##. Show that ##\Phi_1##, ##\Phi_2## and ##\Phi## can be considered as the sides of a spherical triangle with the angle opposite to ##\Phi## given by the angle between the two axes of rotation.

(Source: Classical Mechanics, 3rd edition, Goldstein, Problem 13, Chapter 4)

Homework Equations


If ##A## is a rotation matrix,
$$Tr~A=1+ 2\cos{\theta}$$
,where ##\theta## is the rotation angle.

The Attempt at a Solution


Let ##R_1## and ##A_1## be the rotation axis (unit vector) and the rotation matrix respectively for the ##\Phi_1## rotation. So we get,
$$A_1R_1=R_1$$
and $$Tr~ A_1 = 1+2 \cos{\Phi_1}$$
In the same way, for ##\Phi_2##,
$$A_2R_2=R_2$$
and $$Tr~ A_2 = 1+2 \cos{\Phi_2}$$
The rotation matrix corresponding to ##\Phi## rotation will be ##A_2A_1##. So we get
$$Tr ~A_2A_1 = 1+2 \cos{\Phi}$$
If the angle between ##R_1## and ##R_2## is ##\psi##, then
$$\cos{\psi}=R_1^TR_2$$
Now I need to show that ##\psi## is the angle opposite to ##\Phi## in the spherical triangle with ##\Phi_1##, ##\Phi_2## and ##\Phi## considered as the sides.
I am not sure if the problem can be solved in this approach.
Any help or suggestion will be appreciated.
 
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  • #2
I'm not sure how rigorous the derivation needs to be. But if you construct a diagram of the spherical triangle, then the result appears to follow immediately by inspection of the diagram.
 
  • #3
TSny said:
I'm not sure how rigorous the derivation needs to be. But if you construct a diagram of the spherical triangle, then the result appears to follow immediately by inspection of the diagram.
I was looking for a rigorous derivation.
 
  • #4
OK. I don't see a rigorous derivation at the moment. For what it's worth, here's the diagram that seems to me to show the result.

upload_2017-9-22_23-48-0.png
Start with a spherical apple. Let a radial line sweep out the arc Φ1 from a to b, slicing the apple along the yellow plane. Continue with two more slicings Φ2 and Φ along the blue and green planes, respectively. Remove the wedge of apple that has been sliced out. The picture above shows peering down inside the apple. The red angle is the angle opposite Φ. This angle is clearly the angle between the yellow and blue planes. The result follows by considering how the angle between the yellow and blue planes is related to the angle between the rotation axes corresponding to Φ1 and Φ2.

Hopefully, someone can provide some hints on constructing a rigorous argument.
 
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1. What is the concept of two successive rotation in Goldstein problem 4.13?

The concept of two successive rotation refers to a problem in classical mechanics where a rigid body undergoes two consecutive rotations about different axes. This problem is discussed in detail in problem 4.13 in the book "Classical Mechanics" by Herbert Goldstein.

2. What is the significance of Goldstein problem 4.13?

Goldstein problem 4.13 is significant because it helps in understanding the complex motion of rigid bodies and the effects of multiple rotations on their behavior. It also provides a practical application of the concepts of angular velocity and inertia tensor.

3. Can you explain the steps to solve Goldstein problem 4.13?

To solve Goldstein problem 4.13, the first step is to determine the angular velocity vector for each rotation. Then, the inertia tensor for the body is calculated for each axis of rotation. The two inertia tensors are then combined to find the total inertia tensor for the successive rotations. Finally, the equations of motion are solved using the total inertia tensor to obtain the final solution.

4. How does Goldstein problem 4.13 relate to real-world applications?

Goldstein problem 4.13 has applications in various fields such as aerospace engineering, robotics, and computer graphics. It helps in understanding the motion of objects in 3D space and is crucial in the design and control of spacecraft, satellites, and other mechanical systems.

5. Are there any limitations to the concept of two successive rotation in Goldstein problem 4.13?

One limitation of the concept of two successive rotation is that it assumes the rigid body to be perfectly rigid, which may not be the case in real-world scenarios. Additionally, it does not take into account any external forces acting on the body, which may affect its motion. Therefore, the results obtained from solving Goldstein problem 4.13 may not always accurately reflect the behavior of real-world systems.

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