Doubt in the proof of Euler's rotation theorem

Question

The question arises the way Goldstein proves Euler theorem (3rd Ed pg 156 ) which says:

" In three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point".

The Matrix proof essentially takes an arbitrary $3 \times 3$ orthogonal matrix with real entries and shows that there is at least one vector $n\neq 0$ with $A n=n$ that is an eigenvector with +1 as its eigenvalue .

The author states that this proves the Eulers theorem, which I am not sure why this is true. It seems that all we have shown is that are some vectors that are invariant under $A$ along some line. That doesn't necessarily mean $A$ is a rotation along that line.

Answer 1

Let $A$ be "a displacement of a rigid body such that a point on the rigid body remains fixed", and let $B$ be "a single rotation about some axis." Let's assume we are in three dimensions.

To prove $A$ and $B$ are equivalent, we have to show that $A$ implies $B$ and $B$ implies $A$.

The matrix proof you are describing shows that $B$ implies $A$. In other words, a general rotation (described by an orthogonal matrix) has an axis on which points are invariant under the rotation.

What's left is to show that $A$ implies $B$. At a physicist level of rigor, I think it's fairly obvious that the only way to move a rigid object with one point fixed, is to rotate the object around an axis passing through that point, so at a physicist level of rigor I would consider that proven. Perhaps Goldstein has a more rigorous justification. One way to justify it would be to formalize the idea that a rigid body has 6 degrees of freedom (3 to give the position of one point, and 3 to define its orientation in space). Fixing a point removes three of the degrees of freedom. You can then use 2 of the remaining degrees of freedom to specify an axis, and 1 to specify an angle of rotation about that axis.