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I'm not sure I understand this topic so I would be happy if someone could clarify it for me:

Is precession (say, of a spinning top) generally speaking a change in the direction of the angular velocity vector over time? Is that why it is sometimes said that whenever $\vec{\omega}||\vec{L}$ (angular velocity parallel to the angular momentum) then there is no precession considering there is no torque on the system? In the classical mechanics Landau and Lifshitz book they say that $\omega_p$ (the "precessing" component of the angular velocity) is (when there is no torque), the projection of $\vec{\omega}$ onto $\vec{L}$, why is that? In that case I see that precession is described as the axis of symmetry of a body tracing a circle around the angular momentum vector. In other words, I fail to see how and why $\vec{L}$ and $\vec{\omega}$ both define whenever there is precession and whenever there isn't.

Thank you!

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If a symmetrical object is spinning with an angular velocity, ω, then the angular momentum, L = I ω (with both vectors along the axis of rotation). If an external torque (often from gravity) is applied, then the angular momentum vector (and the axis of rotation) swing in the direction of the torque vector: τ = dL/dt.

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