Classic spinning symmetric top

1. The problem statement, all variables and given/known data
Derive Euler’s equation of motion for a rigid body: $$\dot{\vec{L}} + \vec{\omega} \times \vec{L} = \vec{G},$$ where ##\vec{L}## is the angular momentum in the body frame, ##\omega## is the instantaneous velocity of the body’s rotation and ##\vec{G}## is the external torque.

Subsequent parts are written in an attachment.

2. Relevant equations
Expressions for angular momentum and kinetic energy in rotating frame or body frame fixed in the body.

3. The attempt at a solution
I believe that equation is saying the rate of change of angular momentum in some inertial frame is the rate of change of angular momentum in the body frame + another term due to rotation of axes relative to the inertial frame. Is that right? I found a derivation of the result for a general vector A but I could not understand this equation fully: $$[\dot{A}]_{S_o} = [\dot{A}]_S + \omega \times [A]_S$$ (##S_o## denotes the inertial frame and S the non inertial one and they coincide instantaneously. ##[A]_x## denotes A measured in frame x.)

The term on the left is the rate of change of A in the inertial frame. The first term on the RHS is the rate of change of A in S. Why is it then ##\omega \times [A]_S##? If we consider a turntable of radius R, then the velocity of a point on the circumference is ω x R relative to the centre of the turntable. But the centre of the turntable is fixed (so inertial) and a frame co-moving with a point on the circumference rotates relative to the centre so should it not be ##\omega \times [A]_{S_o}?##
Many thanks.

Attached Images
File Type: jpg Rigid.jpg (76.7 KB)

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