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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?

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The theorem does not say that the actual axis of rotation is fixed. It says that the final configuration can be obtained by a rotation about a single axis. For instance, think about a sphere with its center fixed. Imagine the most general finite motion of this sphere. It will be a composition of many small rotations about different axis. Equivalently, the final configuration can be obtained by a single finite rotation about a fixed axis through its center. This was expected if we recall that rotations form a group. Thus the composition of several rotations is equal to a single rotation.

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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 ! enter image description here

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