Query on the Euler Theorem for Rigid Body Rotation

In summary, Euler's theorem states that any displacement of a rigid body with a fixed point is equivalent to a rotation about some axis. This is proven by showing that an orthogonal matrix must have an eigenvalue of +1 for a proper rotation. However, this does not necessarily prove the equivalence between a rotation and any arbitrary displacement. Additionally, while Euler's theorem states that a rotation can be specified by a single rotation about an axis, three Euler angles are needed to fully specify the orientation of the body because it takes two angles to determine the orientation of the axis itself.
  • #1
Shan K
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Hi,
I am having some problems conceptualizing the Euler's Theorem. Any help will be greatly appreciated.
In Goldstein's book the Euler's theorem is stated as 'Any displacement of a rigid body, whose one point remains fixed throughout, is a rotation about some axis', then he has proven that the orthogonal matrix must have an Eigen Value of +1 for a proper rotation.
1. My question is how does this proves the theorem ?
I have understood the logic behind the +1 eigen value, but could not able to find any equivalence between 'Any Displacement' and 'The rotation'.
2. Another one is that, why should we need three Euler angles for the orientation of a body because from Euler's theorem it can be obtained by only one rotation about some axis ?
Thanks
 
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  • #2
In answer to your second question, Euler's theorem says that any rotation can be specified by a single rotation about some axis. But it takes two angles to specify the orientation of that 'some axis' relative to your coordinate system axes, so three angles in total are required.
 
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Likes Shan K
  • #3
Shan K said:
Hi,
I am having some problems conceptualizing the Euler's Theorem. Any help will be greatly appreciated.
In Goldstein's book the Euler's theorem is stated as 'Any displacement of a rigid body, whose one point remains fixed throughout, is a rotation about some axis', then he has proven that the orthogonal matrix must have an Eigen Value of +1 for a proper rotation.
1. My question is how does this proves the theorem ?
I have understood the logic behind the +1 eigen value, but could not able to find any equivalence between 'Any Displacement' and 'The rotation'.
2. Another one is that, why should we need three Euler angles for the orientation of a body because from Euler's theorem it can be obtained by only one rotation about some axis ?
Thanks
I can try and answer part 1. I hope I understood your question. Here is the argument:
Once you agree that the matrix has an eigenvalue of +1, that means that there is an eigenvector, which is invariant under that transformation. That is precisely what you mean by a rotation about an axis. Any vector along that axis is unchanged by the transformation.
 
  • #4
Chandra Prayaga said:
I can try and answer part 1. I hope I understood your question. Here is the argument:
Once you agree that the matrix has an eigenvalue of +1, that means that there is an eigenvector, which is invariant under that transformation. That is precisely what you mean by a rotation about an axis. Any vector along that axis is unchanged by the transformation.
Thanks Chandra Prayaga for your reply. I have understood this concept but it doesn't prove the equivalence between 'Rotation' and 'any displacement' because in Goldstein they have assumed this equivalence first and then proved that for a rotation there must be an axis, through the proof of +1 eigen value.
They have not proved that this +1 eigen value corresponds to that arbitrary displacement.
Thanks
 
  • #5
phyzguy said:
In answer to your second question, Euler's theorem says that any rotation can be specified by a single rotation about some axis. But it takes two angles to specify the orientation of that 'some axis' relative to your coordinate system axes, so three angles in total are required.
Thanks Phyzguy.
 

1. What is the Euler Theorem for Rigid Body Rotation?

The Euler Theorem for Rigid Body Rotation is a mathematical principle that describes the motion of a rigid body in three-dimensional space. It states that any rigid body motion can be represented by a combination of a rotation around an axis and a translation in the direction of that axis.

2. How is the Euler Theorem used in physics and engineering?

The Euler Theorem is used in physics and engineering to study the motion and dynamics of rigid bodies. It is particularly useful in analyzing the rotation of objects such as spacecraft, airplanes, and rotating machinery.

3. What is the difference between the Euler Theorem and the Euler Angles?

The Euler Theorem and the Euler Angles are two different concepts. The Euler Angles are a set of three angles that describe the orientation of a rigid body in three-dimensional space, while the Euler Theorem is a mathematical principle that relates the motion of a rigid body to these angles.

4. Can the Euler Theorem be applied to non-rigid bodies?

No, the Euler Theorem is only applicable to rigid bodies, which are objects that do not deform under external forces. For non-rigid bodies, other principles such as the Principle of Virtual Work or Lagrange's equations are used to describe their motion.

5. Are there any limitations or assumptions associated with the Euler Theorem?

Yes, the Euler Theorem makes several assumptions, such as the rigid body being in a state of pure rotation or translation, and the absence of external forces. Additionally, it only applies to three-dimensional motion and cannot be used for objects that undergo large deformations.

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