Query on the Euler Theorem for Rigid Body Rotation

[Shan K]

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

 

[phyzguy]

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.

 

[Chandra Prayaga]

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.

 

[Shan K]

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

 

[Shan K]

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.