Precession - why does the rotational axis change as L changes?

[SchroedingersLion]

Hi guys,

while going through rotational energy levels of molecules, I found myself struggling with a very basic concept.
The concept of precession, in terms of a torque induced rotation of the rotational axis of a spinning rigid body.

Take a glance at the Wikipedia figure:
https://en.wikipedia.org/wiki/Precession#/media/File:PrecessionOfATop.svg

Now suppose that the spinning top has the same position as in the figure, only with 0 gravity.
In 0 gravity, it will keep spinning with angular momentum L for all eternity and will not fall down or perform precession or anything.
Now let's turn on the gravity. This leads to a force on the centre of mass pointing downwards, creating a torque M. This torque leads to a change in angular momentum dL=M*dt with dL being a vector standing orhtogonal on L and pointing in a direction tangential to the circular motion of the top surface at its lowest point.
So L --> L+dL.
What I don't really understand: Who says that the axis of rotation of the top now follows the change in vector L?
If that is the case, everything can keep repeating itself and we get precession.
But I don't see that the top axis necessarily follows the change in L. Can't there be some other physical behaviour?
Or, to put it another way, if there is a body with an L vector given, can I trace back this specific L vector to a well defined and unique rotation? Or are there other possible rotational behaviours possible with the same L?

I am a bit confused.

Regards

 

[FactChecker]

Good point. The axis doesn't exactly follow the angular momentum, L. L is the combination of the rotation around the axis and the rotation of the precession.

 

[A.T.]

Who says that the axis of rotation of the top now follows the change in vector L?

What do you mean by "axis of rotation of the top"? The symmetry axis of its geometry, or the actual kinematic axis?

 

[SchroedingersLion]

Good point. The axis doesn't exactly follow the angular momentum, L. L is the combination of the rotation around the axis and the rotation of the precession.

But it still looks like it?

What do you mean by "axis of rotation of the top"? The symmetry axis of its geometry, or the actual kinematic axis?

Yes, the symmetry axis, the axis the top would spin around if its surface was parallel to the ground.

 

[A.T.]

But it still looks like it?

Because the precession is much slower than the spin around the symmetry axis.

 

[FactChecker]

But it still looks like it?

Yes, because usually the rotation around the axis of the top is much faster than the precession. The vector, L, is slightly below the axis of the top. The two rotations combined look exactly as you see it: a fast spinning top slowly precessing.

 

[Dale]

if there is a body with an L vector given, can I trace back this specific L vector to a well defined and unique rotation?

This is an excellent question that I had not considered. I have no reference or proof, but it does seem possible that there is a unique mapping. The L vector has three independent components, and the object can rotate in three motions (spin, precession, nutation)

 

[SchroedingersLion]

This is an excellent question that I had not considered. I have no reference or proof, but it does seem possible that there is a unique mapping. The L vector has three independent components, and the object can rotate in three motions (spin, precession, nutation)

So in general, as long as the L vector does not coincide with a symmetry axis, it is not easy to figure out how the body will rotate?

Thinking about this topic brought to light deeper problems in my understanding.
Think about a rigid cylinder with two short symmetry axes, one of them called SA, and one long symmetry axis LA.
Now, if I had a momentum vector L that coincides with SA, the rotation would be around SA, it would look like a windmill. If L ran along LA I would get spinning around LA, it would look like an advertisement kiosk.
Now what if L pointed along an axis that lies 45° between LA and SA.
Possibility 1) The body would rotate around this axis.
Possibility 2) Since L is a vector, we can decompose it into a component along LA and a component along SA. That would mean that we have the windmill again (rotation around SA) and at the same time, the cylinder would rotate around its LA as with the advertisement kiosk.

But as far as I can tell, the motions of 1) and 2) are not the same. In possibility 1, the condition that all points must maintain the same distance to the rotation axis would imply that some spinning around LA must occur, but I don't see it making a complete spin around one of the SA.
I tried to give my own answer, but I am not sure.
Indeed, 1) and 2) have to be the same. The problem here is that the vector decomposition into the components of 2) is not stationary. For example, as soon as an infinitesimal rotation arround SA occured, the axis LA will have changed its orientation in the lab inertial system. That means that the two components of the L vector change (periodically) in time. At each point in time, the components of L along SA and LA lead to an infinitesimal rotation around SA and LA respectively, and they add up to give an infinitesimal rotation around the 45° axis of the total L vector. But as the orientations of SA and LA change, it is possible that the cylinder makes no complete 360° spin around its SA.

What do you think?

 

[FactChecker]

A given L vector has only one direction and magnitude. That represents one axis of rotation and rotation rate. It may look strange when one sees it in the coordinate system of the gyroscope body, but it is well defined. It's decomposition in the gyroscope body axis is also well defined at any point in time.