Timeline for Motion in the body-fixed frame?
Current License: CC BY-SA 3.0
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| Jul 25, 2012 at 21:35 | comment | added | Art Brown | @gilonik, right, a body-fixed observer might think the body was not spinning (if the surroundings were ignored). Crucially, however, the body-fixed frame is not an inertial frame, so motion is complicated by Coriolis forces. That effect is described by (dG/dt)s=(dG/dt)b + wxG, relating the rate of change in a vector G in the space and body frames, with the angular momentum vector w as defined above. (In that sense, w quantifies the non-inertial-ness.) Setting G=L=Iw, one gets the Euler equations. It's only with that definition of w that one deduces the correct equations of motion. | |
| Jul 25, 2012 at 20:34 | comment | added | gilonik | That helps, thanks. The resolution must be in the instantaneous/infinitesimal caveat. So maybe I just can't picture it. Here's the point: why isn't the body-fixed frame the "rotational rest frame"? I know it isn't: The free precession of the earth shows that we in the body-fixed frame observe an $\vec{L}$ even though we're rotating with the earth. That this $\vec{L}$ differs in magnitude (and direction?) from that observed in the space-fixed frame is fine; that it's non-zero is where I'm getting mixed up. | |
| Jul 25, 2012 at 19:23 | comment | added | Art Brown | @gilonik, I think my answer was a bit too concise, sorry. The body frame is indeed non-inertial. However, to calculate angular velocity, one first establishes an inertial frame that coincides with the body frame at a particular time, and then determines the infinitesimal rotation of the body frame with respect to the inertial frame in a time dt. The angular velocity is that infinitesimal rotation / dt. A reference is Goldstein, Classical Mechanics, section 4-9. | |
| Jul 25, 2012 at 17:10 | comment | added | gilonik | But doesn't that make the body frame inertial -- and isn't the whole point of the Euler equations that they take into account non-inertial "psuedo-torques"? An observer on a spinning object is certainly non-inertial. And in fact, the precession of a free symmetric top is observable by an observer in the body frame: earth's $\omega$ precesses in cone once every 300-400 days. | |
| Jul 24, 2012 at 23:16 | history | answered | Art Brown | CC BY-SA 3.0 |