In a classical mechanics lecture (which I audited) Prof. Poppitz made the claim that an infinitesimal rotation in direction of magnitude has the form
I believe he expressed things in terms of the differential displacement
This was verified for the special case and . Let’s derive this in the general case too.
With geometric algebra.
Let’s temporarily dispense with the normal notation and introduce two perpendicular unit vectors , and in the plane of the rotation. Relate these to the unit normal with
A rotation through an angle (infinitesimal or otherwise) is then
Suppose that we decompose into components in the plane and in the direction of the normal . We have
The exponentials commute with the vector, and anticommute otherwise, leaving us with
In the last line we use and . Making the angle infinitesimal we have
We have only to confirm that this matches the assumed cross product representation
Taking the two last computations we find
Without geometric algebra.
We’ve also done the setup above to verify this result without GA. Here we wish to apply the rotation to the coordinate vector of in the basis which gives us
But as we’ve shown, this last coordinate vector is just , and we get our desired result using plain old fashioned matrix algebra as well.
Really the only difference between this and what was done in class is that there’s no assumption here that .