The vector differential equation $\frac{d L}{d t}=v \times L$ mathematically describes pure precession, where the angular momentum vector $L$ rotates about the fixed axis defined by the constant vector $v$. The two fundamental takeaways are derived by showing that the time derivatives of $|L|^2$ and $v \cdot L$ are both zero, relying on the property that the cross product result $v \times L$ is always perpendicular to both $v$ and $L$. This proof confirms that the magnitude of $L$ is conserved, and its component parallel to $v$ is also conserved. Geometrically, this means $L$ traces a cone of constant apex angle and constant radius around the constant vector $v$, indicating that the rotation is a steady, non-diminishing precession, typical of systems like spinning tops or magnetic spins in a uniform field.
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