It seems that the universally accepted interpretation of the Dzhanibekov effect is that it's nothing but a case of the intermediate axis theorem, which has been known in classical mechanics for at least 150 years.

The tennis racket effect is verified easily with simple experiments, so there shall be no doubt in its validity. However, Dzhanibekov effect cannot be reduced to the intermediate axis theorem – a point that seems has not been pointed out yet.

To be specific, it will be demonstrated below that there are cases of the Dzhanibekov effect which clearly contradict the intermediate axis theorem. (The theorem states: rotation of a rigid body with three distinct principal moments of inertia, $I_1 > I_2 > I_3$, about the intermediate axis is unstable, while rotation about two others is stable).

Now consider the following two axisymmetric cases:

(1) $I_1 = I_2 > I_3$ with a spin about the axis of least moment of inertia;

(2) $I_1 > I_2 = I_3$ with a spin about the axis of maximal moment of inertia.

The intermediate axis theorem has nothing to say about these borderline cases. But it can be proved directly that spin in both cases is still stable, so there shall be no flipping of the body. Call it an extension of the intermediate axis theorem, if you wish.

**Proof:** In *torque-free conditions*, the equations of rotation of asymmetric top around its center of inertia in the frame of reference which is rigidly fixed to the body, axes $x_1, x_2, x_3$ being directed along the body’s three principal axes of inertia, are:

$$
I_1\dot{\omega}_1 = (I_2 - I_3) \omega_2\omega_3,
$$
$$
I_2\dot{\omega}_2 = (I_3 - I_1) \omega_3\omega_1,
$$
$$
I_3\dot{\omega}_3 = (I_1 - I_2) \omega_1\omega_2.
$$

Here $I_1, I_2, I_3$ denote the body's principal moments of inertia while the angular velocities around the corresponding axes are denoted by $\omega_1, \omega _2, \omega _3$.

In the case of axisymmetric top, say $I_1 = I_2$ (both being equal to some $I$), we have Euler’s equations reduced to:

$$
I\dot{\omega}_1 = (I - I_3)\omega_2\omega_3,
$$
$$
I\dot{\omega}_2 = (I_3 - I)\omega_3\omega_1,
$$
$$
I_3\dot{\omega}_3 = 0.
$$

From the third equation we have $\omega_3 =$ const, so the other two are greatly simplified:

$$
\dot{\omega}_1 = +\Omega \omega_2,
$$
$$
\dot{\omega}_2 = -\Omega \omega_1,
$$

where $\Omega \equiv (I - I_3)\omega_3/I$.

Therefore, we have an analytical solution with a fixed angular velocity of precession $\Omega$, which prevents the axis of rotation from flipping over:

$$
\omega_1(t) = \omega \sin(\Omega t),
$$
$$
\omega_2(t) = \omega \cos(\Omega t),
$$
$$
\omega_3(t) = \omega_3 = \text{const}.
$$

The constants of integration $\omega$ and $\omega_3$ are determined uniquely by the angular momentum $M$ and the angle of precession $\theta$:
$$
\omega = M\sin(\theta)/I,
$$
$$
\omega_3 = M\cos(\theta)/I_3.
$$
**End of proof**.

Now we proceed to the key point: Dzhanibekov has observed flipping not only for a wing nut (asymmetric object with $I_1 > I_2 > I_3$ spinning about the intermediate axis), which is consistent with the tennis racket theorem, but also for a nearly axisymmetric body (a regular hexagon nut attached to a ball of modelling clay $-$ a body with $I_1 \gtrapprox I_2 > I_3$ spinning about the least-inertia axis) in clear violation of the intermediate axis theorem:

The Bizarre Behavior of Rotating Bodies by Veritasium.

(Note that the density of iron is higher than the density of plasticine, so there can be no doubt that we have here an object with $I_1 \gtrapprox I_2 > I_3$ spinning about the least-inertia axis. For brevity, let us call this case *the Dzhanibekov top*.)

**Conclusion:** Equating Dzhanibekov effect with the tennis racket instability is a blunder. So, the real physical cause for the instability of the Dzhanibekov top needs to be identified.

It is certain that dissipation of kinetic energy, which is due to *internal* forces, cannot be the physical cause of this effect. Indeed, even if we allow for this mechanism, all it can do is to increase the angle of precession to its maximum possible value of $\theta = \pi/2$, at which point the dissipation of kinetic energy must stop. So, the flipping back and forth of the Dzhanibekov top cannot be accounted for by the energy dissipation mechanism.

Therefore remains only one credible explanation for the behavior of the Dzhanibekov top: it is due to an external torque, aerodynamic interaction with the surrounding air being the most likely candidate for the source of the external torque. This can be tested easily. If the Dzhanibekov top stops flipping in a vacuum (say, outside the ISS), we can be certain that aerodynamic interaction with the air inside the ISS was the real culprit for the effect.

From the viewpoint of the physical cause of flipping, the Dzhanibekov top is a cousin of the Thomson top (tippe top), rather than the tennis racket. Indeed, flipping of the tennis racket will persist in vacuum conditions; on the other hand, flipping of the Dzhanibekov top requires the assistance of external torques, that is – just like the tippe top – it cannot take place in a torque-free environment.

**P.S.** Take a look at this picture of Wolfgang Pauli and Niels Bohr – two old guys spellbound and observing with childish elation the weird pattern of motion of a fast spinning tippe top:

http://www.fysikbasen.dk/Images/Figurer/pauli_bohr_tippetop.jpg

Isn't it lovely?

The picture is taken at the opening of the new institute of physics at the University of Lund on May 31 1951. Credit: Photograph by Erik Gustafson, courtesy AIP Emilio Segre Visual Archives, Margrethe Bohr Collection.

self-sufficientanswer explaining this wonderful phenomenon! $\endgroup$ – Suvrit Nov 27 '11 at 12:214more comments