Stability of Rotation
Alexei Gilchrist
1 Background
Starting from the equation for torque-free motion:
2 Small Changes
Try the following experiment: take a hardcover book held closed by rubber bands, and try to throw it so it spins around each of the principal axes. First of all, notice that it’s really hard to set up the spin exactly on axis. Secondly, you will find that it’s easy to spin the book with a small off-axis wobble around \(\hat{x}_1\) and \(\hat{x}_3\), but not \(\hat{x}_2\) (where \(I_1>I_2>I_3\)).
Let’s analyse the situation in the example, where the object initially starts to spin with \(\vec{\omega}\) close to one of the principal axes.
2.1 Spin around \(\hat{x}_1\)
If it’s spinning nearly around \(\hat{x}_1\), this means \(\omega_1\gg\omega_2,\omega_3\), and so \(\omega_2\) and \(\omega_3\) are tiny. It follows from \eqref{eq:torquefreepositive} that \(\dot{\omega}_1 = A \omega_2 \omega_3\approx 0\) since it’s quadratic in a small quantities. This means we can treat \(\omega_1\) as constant, at least to a first approximation.
The remaining two equations are

An alternative way of deriving this result is to differentiate the equations once more:
2.2 Spin around \(\hat{x}_3\)
Spinning the object around \(\hat{x}_3\) is very similar to spinning it around \(\hat{x}_1\). In fact, to derive the equations of motion we just need to make the substitutions \(\omega_1 \leftrightarrow \omega_3\), \(A \leftrightarrow C\), to arrive at
2.3 Spin around \(\hat{x}_2\)
Whereas in the previous two cases if we start the object spinning near \(\hat{x}_1\) or \(\hat{x}_3\) we will remain spinning near those axes. The axis \(\hat{x}_2\) is different, and the difference arises only because of a change of sign of a term. Proceeding along the same steps as in the previous cases we arrive at
This is why it’s nearly impossible to spin an object around the intermediate principal axis. Any slight deviation away from the axis get’s magnified, and very quickly the assumption that \(\dot{\omega}_2 \approx 0\) is no longer valid and much more complex motion ensues.
3 Energy and Angular Momentum Conservation
There are no torques acting on the system so that the magnitude of the angular momentum, \(L\), must be constant in time. This means that
Now, the total energy is also conserved, and since it’s a freely spinning object, this is entirely kinetic energy \(T\). The kinetic energy due to rotation is

Both conservation laws hold simultaneously so the \(\vec{L}\) vector has to lie on both surfaces simultaneously, or in other words at the intersection of the two surfaces. The simulation below show both surfaces and their intersection. Initially \(I_1>I_2>I_3\), and by changing \(L\) and \(T\) you can find solutions that stay close to \(\hat{x}_1\) or \(\hat{x}_3\) but not \(\hat{x}_2\)—the surfaces just don’t intersect in a favourable way to allow \(\vec{L}\) to remain near \(\hat{x}_2\). This is of course a graphical confirmation of what we found in the small changes section.
Instead of dynamically investigating the intersection of the two surfaces, we can summarise the type of orbits available by plotting angular momentum contours on the energy ellipsoid. Again pay particular attention to the orbits near each of the three principal axes.

4 Damping
If the spinning body is not quite isolated from the environment then the conservation laws we have been using no longer hold strictly. What will happen to the motion? Imagine that the body dissipates energy through some microscopic motion. This could arise for instance if the object can deform slightly in response to the stresses of its rotation. Those small shifts and motions would leak away energy from the macroscopic degrees of freedom and be lost as heat. That microscopic motion is random as far as the object is concerned so it will not set up a net torque. Consequently, we expect energy to be lost from the system but the angular momentum will be preserved.
As we found above, the total energy is purely kinetic for a free-spinning object and is given by
The effect can be summarised glibly as flying saucers make better spacecraft than flying cigars!
In 1958 United States launched it’s first satellite, Explorer I. This was the year after the Soviet Union launched Sputnik I so there was somewhat tense competition between the two nations. The instrumentation was designed by van Allen to explore the nature of cosmic rays in Earth’s orbit and it discovered evidence for radiation belts which have since become know as Van Allen Belts.

Explorer I was designed to spin about its long axis but soon after launch the satellite mysteriously started to precess. Amongst its parts the satellite had some flexible antennae `whiskers’ ...
As you can imagine, these elements dissipated energy without introducing a torque and the satellite re-orientated itself to spin about the axis with highest moment of inertia.