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*From*: John Denker <jsd@av8n.com>*Date*: Wed, 26 Dec 2012 19:41:09 -0700

On 12/26/2012 07:26 AM, Carl Mungan wrote:

I'm looking for an intuitive argument for the intermediate-axis

theorem (IAT), rather than a formal derivation via Euler's

equations.

Gack! In my previous answer, I violated an important principle:

When talking about stability, it pays to say explicitly "stable

with respect to WHAT?"

Web searches indicate that I am not the first person to make

this mistake in the context of this problem.

So, there are two separate versions of this problem to be

considered ... plus an interesting interaction between the

two versions.

A) The easy version applies to an isolated _almost_ rigid

system. There is a tiny bit of internal dissipation. The

dissipation arises from the fact that the system is not 100%

rigid.

The assumptions guarantee that the system has constant

angular momentum ... but not necessarily constant rotational

energy. Some of the energy might be dissipated as heat.

This can be analyzed in terms of basic 3rd-grade physics.

As surely as balls roll downhill and are stable at the

bottom of the hill, this system is stable (in the

thermodynamic sense) when rotating around the axis with

the largest angular momentum.

B) The more challenging case, which was probably the one CM

intended, assumes constant rotational energy as well as

constant angular momentum.

Here we are interested in stability in the Lyapunov sense.

That is, given nearby initial conditions, do we get nearby

trajectories in the long term?

The intuitive way to handle this is to think about it in

angular momentum space. The contours of constant energy

are ellipsoids in this space. Let's choose one such

contour, i.e. let's choose a definite energy.

If the norm of the angular momentum is as small as possible

(consistent with the chosen energy) the angular momentum must

be at the short axis of the ellipsoid. If the norm is merely

/near/ the smallest possible value, the angular momentum must

be near the short axis. The geometry of spheres intersecting

with an ellipsoid guarantees it.

Similarly, if the norm of the angular momentum is at or near

the largest possible value, the angular momentum vector must

be at or near the long axis of the ellipsoid.

Now (!) consider the intermediate case. There are lots of

places the angular momentum vector could wander relative to

the ellipsoid. (In any inertial frame the angular momentum

vector doesn't wander at all, but the ellipsoid does, as

we re-orient the object. Alternatively, in a frame comoving

with the body, the ellipsoid stays fixed but the angular

momentum wanders.)

So, in the intermediate case I haven't yet proved that it

will tumble, but I have explained why it could tumble (and

could not tumble in the other two cases). Proving that it

*will* tumble might require a more detailed analysis.

I tried googling for images of the intersection of spheres

with an ellipsoid. Here is the crucial intermediate case:

http://api.ning.com/files/50GrcSOUEsM8m6GLfnIFLQe5jrT5bi-VvI51otk7sHFl*xzmqT8NEvugFn2MEdwAMjqW9eFeom3F0KQu161tQaDyJoqt*Ero/CirclesOfEllipsoid.png

It should be trivial to cobble up more such images using x3d

aka VRML. I can easily visualize such things in my mind's

eye, but when explaining it to somebody else it helps to

have the picture.

C) In the real world, the object always has some internal

friction. If the object is spinning on the "big" axis, it

is unconditionally stable, so the friction doesn't matter.

Interestingly, if it is spinning smoothly on the "small"

axis, the friction might act only slowly. This axis is

therefore in some sense metastable. In the intermediate

case, the system is wobbling like crazy, causing lots of

internal friction, so it will rather quickly decay to the

lowest-energy state.

This case (C) is a weird combination of the previous two

cases.

**Follow-Ups**:**Re: [Phys-L] intermediate axis theorem***From:*Carl Mungan <mungan@usna.edu>

**References**:**[Phys-L] intermediate axis theorem***From:*Carl Mungan <mungan@usna.edu>

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