Link to the MathOverflow question: http://mathoverflow.net/questions/81960/dzhanibekov-effect-exercise-on-mechanics-or-a-fiction-explain-this-http

### Terence Tao

Shared publicly -The tennis racket theorem, demonstrated in a Soviet space station. (Via Math Overflow.)

(The tennis racket theorem asserts that when rotating a rigid body with three distinct moments of inertia, the rotation around the axes with the largest or smallest moments of inertia is stable, but the rotation around the axis with the intermediate moment of inertia is unstable. Indeed, in the latter case the object will (when one looks just at the angular velocities) typically traverse periodically through the space of all states with the given angular momentum and energy, which is a closed curve known as a

(The tennis racket theorem asserts that when rotating a rigid body with three distinct moments of inertia, the rotation around the axes with the largest or smallest moments of inertia is stable, but the rotation around the axis with the intermediate moment of inertia is unstable. Indeed, in the latter case the object will (when one looks just at the angular velocities) typically traverse periodically through the space of all states with the given angular momentum and energy, which is a closed curve known as a

*herpolhode*that will pass close to both antipodes of the unstable equilibrium in an alternating fashion.)107

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Vladimir Nesov

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(Note that the video is from a pseudoscience film, and towards the end of the clip they start going on about the Earth making these leaps every 12K years...)

Terence Tao

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+Tau-Mu Yi Actually, it is more of a bi-instability than a bistability; the object is oscillating between two unstable equilibria, with no stable equilibrium in between.

+Vladimir Nesov Ah, I was wondering about that (I don't speak Russian). Of course, this effect doesn't actually apply to the Earth, which is essentially rotating around the axis with the largest moment of inertia, rather than the median moment of inertia (the Earth is a slightly oblate spheroid that bulges at the equator), so the rotation is stable rather than unstable (i.e. it exhibits only mild precession around the major axis of inertia).

+Vladimir Nesov Ah, I was wondering about that (I don't speak Russian). Of course, this effect doesn't actually apply to the Earth, which is essentially rotating around the axis with the largest moment of inertia, rather than the median moment of inertia (the Earth is a slightly oblate spheroid that bulges at the equator), so the rotation is stable rather than unstable (i.e. it exhibits only mild precession around the major axis of inertia).

Zvika Ben-Haim

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So what would happen in

*n*dimensions? Would all axes except the highest and lowest be unstable?Terence Tao

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+Zvika Ben-Haim Yes; this can be deduced from the 3D case by restricting attention to motions that fix all but three of the axes of inertia. (Or, one can look at the higher-dimensional Euler equations for rigid motion and perform a stability analysis there.)

Terence Tao

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Incidentally, the tennis racket theorem can easily be demonstrated without the need for a space station: Tennis Racket Theorem

Is the potato-looking object fake? Can an object of that shape and that axis of rotation be unstable?

Thank you, amazing.

But constant please.I'll show what I mean while I'm under the bridge.My gmail is not private.If you don't understand ( but constant please ).So don't answer me please.

man, i learned about this 6 years ago

Jayden Newstead

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+Aleksandar Nikolov I believe the 'potato' object has the same winged nut inside it, showing that the phenomena is not related to the object's aerodynamics.

The potato was a bit confusing to me at first. I first assumed it was CGI, then that it was real and launched the same way as the winged nut was. But it would be hard to find material light enough that even at a larger radius, the mass distribution of the nut would dominate. Looking at the linear motion of the thing, it seems that the center of mass is very close to the center of volume, so I'm guessing they embedded some masses in a ball of clay, and left a mass on the outside to mark the axis. Anyway, the potato isn't hollow, so I'm at a loss to how they launched it with the right orientation. Digital film is cheap, I guess.

With my poor Russian (or rather by analogy with my native Bulgarian), I think they say that the "potato" is made of modeling clay. But that is quite heavy, the kind of modeling clay I am familiar with will outweigh the nut at this size.

I also like the bit at 2:35 where the Earth has flipped chirality. The rest of the animations have the continents the right way out, which makes it extra funny.

Scott Carnahan

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I was under the impression that the Earth does exhibit true polar wander, but on a roughly 500Ma time scale rather than a 12Ka time scale - this is enough time for the mass distribution to shift due to plate tectonics and other geologic processes.

As Terry Tao already replied to Vladimir Nesov, the described phenomenon does not apply to the Earth. Totally different is the geomagnetic reversal whose possible dynamical model is the mechanics of the rattleback as proposed by Moffatt and Tokieda in their paper here: http://www.igf.fuw.edu.pl/KB/HKM/PDF/Moffatt_Tokieda_2008.pdf .

We used to do this exercise in my college physics class however the tennis racket was replaced with the textbook for the course and you just had to throw it up in the air hard enough that it would rotate enough times before it fell back down. It helped to put a rubber band around the book and also remember to subtract off the projectile motion of the center of mass (thus the same problem as on the space station).

I have made a machine,what you think the engineering department would do with it if I give it to them?.

+Andrea Bertozzi Indeed, a book with a rubber band around it is a good way to demonstrate the tennis rack theorem. When it is tossed up, if it is set spinning around the axis with the intermediate moment of inertia, it will tend to tumble, and be hard to catch.

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