Precession of Angular momentum of Symmetric Top For a torque free symmetric top, is the angular momentum in body fixed coordinates in same direction as instantaneous axis of rotation?
I know that instantaneous axis of rotation precesses about symmetry axis, and Goldstein says that the angular momentum of the torque free symmetrical top rotates in body coordinates about the symmetry axis with an angular frequency $\Omega$.
But I know that $\vec{\omega}$ precesses about symmetry axis with angular speed $\Omega$. So, is $\vec{L}$ along instantaneous axis of rotation?
And are there two $\vec{L}$’s, one in body coordinates and another in a fixed spatial frame?
 A: For a torque-free body, the total angular momentum is necessarily a constant of the motion.  If you use a fixed coordinate system $\vec{L}$ will be manifestly constant.  In the body coordinates, which are themselves time dependent, $\vec{L}$ will not have a manifestly time-independent form, precisely because the coordinate axes are moving.
Ultimately, since $\vec{L}$ is a fixed (axial) vector, the most natural description of any other (axial) vector $\vec{A}$ will be in terms of what $\vec{A}$ is doing relative to the fixed direction $\vec{L}$.  The instantaneous rotation vector $\vec{\omega}$ precesses around the fixed direction $\vec{L}$.  The total angular momentum includes contributions both from the spinning of the top at rate $\omega$ and its precession at $\Omega$.  There is a standard description of how everything precesses around $\vec{L}$ in terms of two cones (the space cone and the body cone) rolling against each other, as shown schematically in this figure.  (Which of the two diagrams applies depends on the geometry of the top.)

