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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?

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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.)

space cone/body cone

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  • $\begingroup$ Ok, so angular momentum is a fixed axial vector. Then can you please explain what Goldstein saying(in my question) "the angular momentum of torque free symmetric top rotates in body coordinates...". Is the statement wrong? $\endgroup$
    – Real Gamer
    Aug 13 '20 at 17:15
  • $\begingroup$ @RealGamer If you measure $\vec{L}$ in a frame that is itself precessing (which the body frame is), the components of $\vec{L}$ (measured in the frame) will look like they are moving. Just like from the rotating frame of the Earth, it looks like the sun is moving around us. $\endgroup$
    – Buzz
    Aug 13 '20 at 17:41
  • $\begingroup$ In body coordinates, $\vec{L}$ will be seen moving. According to "the angular momentum of torque free..." $\vec{L}$ rotates about symmetric axis with angular speed $\Omega$ in body coordinates. And we know $\vec\omega$ precesses about symmetry axis with angular speed $\Omega$. Can I say $\vec{L}$ in body coordinates has direction of $\vec\omega$ ? $\endgroup$
    – Real Gamer
    Aug 13 '20 at 17:59
  • $\begingroup$ @RealGamer If two vectors are in the same direction in one frame, that applies in every frame. So if there is precession going on, $\vec{L}$ and $\vec{\omega}$ are not proportional in the body frame or any other frame. $\endgroup$
    – Buzz
    Aug 13 '20 at 18:36
  • $\begingroup$ 1. $\vec{L}$ in body frame rotates about symmetric axis with $\Omega$.2.$\vec\omega$ precesses symmetric axis with $\Omega$. 3. And $\vec{L}$ and $\vec\omega$ are not proportional. Are these conclusions correct? $\endgroup$
    – Real Gamer
    Aug 13 '20 at 18:46

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