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I am totally confused about Euler's rotation theorem.

Normally I would think that an asteroid could rotate around two axes simultaneously. But Euler's rotation theorem states that:

In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about a fixed axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation.

But asteroids do rotate around two axes. Look at the videos from this website: http://csep10.phys.utk.edu/astr161/lect/asteroids/features.html

The Spin of Asteroid Toutatis

By making a series of observations, it is possible to study the rotation of some asteroids. Most have simple rotations around a fixed axis, with periods typically between one hour and one day. For example, here is a movie (83 kB MPEG) made by the Hubble Space Telescope of the asteroid Vesta in rotation (Ref). However, the asteroid 4179 Toutatis (which crosses Earth's orbit) has been found through radio telescope observations to have an irregular shape and a complex tumbling rotation---both thought to arise from a history of violent collisions. Here is a short animation (47 kB MPEG) of the spin of Toutatis; here is a longer animation (288 kB MPEG).

Here are the movies:

http://csep10.phys.utk.edu/astr161/lect/asteroids/toutspin.mpg

and

http://csep10.phys.utk.edu/astr161/lect/asteroids/toutspin2.mpg

You can clearly see that those rotations are not possible around one axis. But then isn't it contradicting Euler's rotation theorem?

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    It seems that all the links in the question do not work anymore. But [here a video](https://youtu.be/4a8pwaIBm-Q) showing the weird rotation of the astroid Toutatis.2015-07-29

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I think you are confusing a single rotation (as a fixed displacement) with a rotational motion.

Euler theorem says that the composition of two individual rotations (say, I rotate a body 15 degrees around a vertical axis, then I rotate it 10 degrees around some horizontal axis) is equivalent to a single rotation around some axis.

But suppose I do the same double rotation, with the same axes but, say, double angles ( 30 and 20 degrees): they would be, again, equivalent to a single rotation, but the equivalent axis would be different.

Hence, the composition of two rotation (motions) with fixed axis is not equivalent to some other rotation (motion) with another (fixed) axis. Then, to speak of the superposition of two rotational motions (that cannot be reduced to a single rotational motion) makes perfect sense.

Example: Take a long cilinder, make it rotate quicky along its longitudinal axis. Superpose to that a slow rotation along a transversal axis. If both axis passes through the center of the cylinder, there is a fixed point. However, the resulting rotational motion cannot be expressed as a single rotational motion (with a fixed axis).

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    There are 3 "fixed" in the original sentence, I think we are talking about different ones. "Remains fixed" and "fixed axis" and "fixed point". Leonboy were talking about "fixed axis" and I think the wiki article is now correct.2011-06-11
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Hmm well actually for any rigid body, it does rotate along an axis. (at least for a infinitesimal amount of time) and this rotation will stay around the same axis over time as long as no torque is applied. It may just be difficult to see that axis.

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    We learned in experimental physics class at the university that a rigid body always can rotate stably around 2 of its 3 axes. The longest axis and the shortest one. The rotation is totally unstable around the middle axis. You can show that by throwing a rotating book up into the air and watch how it performs.2015-01-08