# 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.

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