# Axis of rotation and Euler's theorem in rigid body dynamics

Euler's theorem of Rotation for rigid body states that

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.

Intuitively, the axis of rotation can change from $\hat{\textbf{n}}$ to $\hat{\textbf{n}}^\prime$ from one instant $t$ to the next $t^\prime$ in a continuous manner. But the motion need not be such that there is a finite time during which the system maintains a fixed axis. How is it then meaningful to talk about the axis of rotation at a given instant?

I think the meaning is: at that instant time, you can treat the body like it is rotating around an axis, that is, you can have thing like: the velocity of any point on the rigid body can be obtained by: $\vec v = \vec \omega \space \times\space \vec r$ , of course these values of velocities only hold at that instant time !