Deriving position from angular velocity on a rotating sphere.

[meiji1]

Homework Statement

Consider a rigid sphere of radius 1 and center at (0,0,0) that rotates about its center. The
angular velocity is $\omega(t) = (\cos(t) , \sin(t), \sqrt(3))$. Does the path of the point starting at (0,0,1) ever reach this point at a later time?

Homework Equations

We're given $\omega(t) = (\cos(t) , \sin(t), \sqrt(3))$ as the angular velocity of the rotation of the sphere, and that's it.

The Attempt at a Solution

Let $r(t)$ denote the displacement of (0,0,1) on the sphere at time t, so that $r(0) = (0,0,1)$.

(in this notation, Skew(\omega(t)) := the tensor of \omega(t), I believe)

I know that we can write $dr/dt = Skew(\omega(t)) r(t)$, and from this we induce a linear system of differential equations. From the definition of $\omega(t)$, the matrix elements of $Skew(\omega(t))$ will be non-constant. I have no idea how to solve a linear system of differential equations with non-constant coefficients. Obviously the eigenvector approach used in the constant case would fail horrendously.

Perhaps I should instead ask for a reference on that. If that's too hard or complicated a problem to tackle, could someone please suggest a different approach?

On a broader note.. the various sources I've read on angular velocity tend to say it's always parallel to the axis of rotation, but I don't believe that's true. My textbook does an example in which the angular velocity of a particle rotating about a moving axis is derived, and it is not in general parallel to whatever the axis of rotation happens to be at time t. Nonetheless, the infallible source of knowledge of all things science, Wikipedia, says without qualification that the angular velocity is parallel to the axis of rotation.. very confusing. :-/

 

[Spinnor]

The magnitude of the angular velocity might be important as well as the rate at which the angular velocity vector rotates about the z axis?