classical mechanics - Rotation axis of a rigid body

I am confused about a trivial concept. Let the rotation of a rigid body, say with one point fixed, be described by the equation $\vec{x}(t)=R(t)\vec{x}(0)$, with $R(0)=I$.

Then, at each instant there is only one real eigenvector of $R(t)$ with eigenvalue 1 that we may call $\vec{v}(t)$ and which we may take to be normalized. That vector $\vec{v}(t)$ is what geometrically we would call the (instantaneous) axis of the rotation.

Kinematically, however, the instantaneous axis of rotation is the line of points with vanishing instantaneous velocity $\dot{\vec{x}}(t)=\vec{\omega}(t)\times\vec{x}(t)=0$. That is the direction of $\vec{\omega}(t)$.

As is obvious (for example from the Rodrigues formula), in general $\vec{v}(t)$ and $\vec{\omega}(t)$ are not parallel. So, why are there two axes of rotation, and does $\vec{v}(t)$ play any role in the kinematics/dynamics of the motion?

Answer

I will translate your post into the language of translations. Then I will answer this question about translations. Then I will answer your original question.

I am confused about a trivial concept. Let the displacement of a rigid body be described by the equation $\vec{x}(t)=\Delta \vec{x}(t) +\vec{x}(0)$, with $\Delta \vec{x}(0)=0$.

Then, at each instant there is only one unit vector in the direction of displacement that we may call $\hat{w}(t)$ and which we may take to be normalized. That vector $\hat{w}(t)$ is what geometrically we would call the (instantaneous) direction of velocity [Note: this sentence is wrong].

Kinematically, however, the instantaneous direction of velocity $\vec{v}$ is the derivative $\dot{\vec{x}}(t)=\dot{\Delta \vec{x}}(t)$. That is the direction of $\vec{v}(t)$.

As is obvious (for example an object not moving in a straight line), in general $\hat{w}(t)$ and $\vec{v}(t)$ are not parallel. So, why are there two directions for velocity, and does $\hat{w}(t)$ play any role in the kinematics/dynamics of the motion?

You are wrong that $\hat{w}$ is the direction of velocity. $\hat{w}$ was defined as the direction of $\Delta \vec{x}$, that is, the direction of the displacement. As Kevin said the total displacement is not needed because you only need to know a objects current position and velocity to get its motion in the future.

Now we are ready to answer the rotation question. We just translate the answer of the translation question to rotation language.

You are wrong that $\vec{v}$ is the direction of angular velocity. $\vec{v}$ was defined as the axis of rotation for $R$, that is, the direction of the rotation between the initial and final orientation. As Kevin said this rotation is not needed because you only need to know a objects current position and velocity (here we are talking about the velocity everywhere in the object, which for a rigid object can be summarized by a linear velocity and an angular velocity) to get its motion in the future.