# The rotating movement of an asteroid

I almost all movies where you could see an animation about an asteroid, they move in a very distinct way.

I don't know how to explain better, but I think what we can see in the movies is that the asteroid is rotating around the x axis with constant speed, around the y axis with constant speed and around the z axis with constant speed, where x, y and z axes are of a reference frame.

What is this kind of motion? Is it correct?

According to the Euler's rotation theorem any 3D rotation that has a fixed point also has a fixed axis. So if they are rotating around more then one axis, does it mean that they don't have any fixed point? But then what is the motion what their center of mass is doing?

I would suppose, that the correct movement would be when asteroids are rotating like planets, around one fixed axis. But to me the rotation commonly used is not like that. What is the correct way?

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Euler's rotational theorem only states, that we can determine a unique axis of the rotation for any given moment. That does not necessarily mean that the axis of rotation's direction is fixed forever.

Quite opposite, let's suppose that we have body with no external force acting upon it. If axis of rotation at some (starting) moment does not coincide with one of the three principal axis of moment of inertia, the axis of rotation shall change and you shall have complex rotation of that body (precession). You can see that directly from Euler's equations by putting torque to be zero.

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Any motion of a rigid object can be decomposed into translation (straight-line motion) of the center of mass, and rotation around the center of mass. In a movie scene, an asteroid would probably be both rotating and translating with respect to the camera, because it makes the motion look more complicated that way. But if you imagine a different camera moving along with the asteroid, it would see the motion as only rotation.

Now, any rotation can always be expressed as a rotation around a fixed axis with a fixed speed. Even if it appears that the asteroid is rotating around several different axes simultaneously, there really is just one axis, according to Euler's theorem. In movie scenes, they tend to orient that axis at an odd angle to the camera, again to make it look more complicated, but given enough information about the characteristics of the motion, you can always find the one axis of rotation and identify the speed of the rotation around it.

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The combined motion of the camera and the asteroid sounds interesting, I had not considered that. Can the camera's motion really not be combined like angular momentum vectors? Obviously it can't to the point that it's not perfect circular motion, but assuming it is my curiosity still stands. Good question to take care of btw, this is one of the most common misconceptions of physics I hear from regular people. – Alan Rominger Jun 11 '11 at 2:50
@Zassounotsukushi: well, it's all about relative motion, both the translation and rotation. You certainly could have a camera that not only translates along with the asteroid, but also rotates along with it, so that the asteroid would appear to be stationary. Was that what you were asking about? (If not it might be better to take this up in the chat room.) – David Z Jun 11 '11 at 3:09
I feel this is sufficiently answered now. I hadn't considered camera rotation, which certainly, combined with everything else could give some non-trivial apparent motion. That could be an entire question unto itself. – Alan Rominger Jun 11 '11 at 3:25

According to the Euler's rotation theorem any 3D rotation that has a fixed point also has a fixed axis. So if they are rotating around more then one axis, does it mean that they don't have any fixed point? But then what is the motion what their center of mass is doing?

This is a common confusion. Typically, one of the rotations has an axis that is fixed only with respect to the object itself, not to the reference universe or camera. Hence, Euler theorem does not apply (or only applies instanteously).

Let disregard translations. Imagine a tall vertical cylinder, with its center fixed, which rotates quickly around its main axis (say, Z). Now, superpose to that a slow rotation around some other line (that passes through the fixed center). The resulting motion (typical of those asteroids), is a composition of two rotational motions, but it's evident that it cannot be expressed as a single rotational motion around some fixed axis. Does this violate Euler theorem? No, because the two component rotations have not two fixed axis (with respect to any reference), hence there is not a fixed "total" rotation axis.

But, you might say, if in the above example we consider the position of the object in two different times T1 and T2, can that displacement be expressed as a rotation with a fixed potin? Yes. Does Euler theorem implies that there is some axis for that rotation? Yes. But that axis would be different if we considered different times.

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