Stability of Rotation

1 Background

Starting from the equation for torque-free motion:

which, assuming \(I_1>I_2>I_3\), we will write as where \(A\), \(B\), and \(C\) are positive constants.

2 Small Changes

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

If we introduce the two scaled quantities \(\tilde{\omega}_2 = \sqrt{C}\omega_2\) and \(\tilde{\omega}_3 = \sqrt{B}\omega_3\), these equations can be written as which are the equation for motion in a circle in the \(\omega_3\)-\(\omega_2\) plane with angular frequency \(\sqrt{BC}\omega_1\).

Rotation around axis \(\hat{x}_1\).

An alternative way of deriving this result is to differentiate the equations once more:

and in both cases this describes simple harmonic motion with an angular frequency of \(\sqrt{BC}\omega_1\)—which is the projection of the circular motion onto the \(\omega_2\) and \(\omega_3\) axes.

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

where \(\tilde{\omega}_2 = \sqrt{A}\omega_2\) and \(\tilde{\omega}_1 = \sqrt{B}\omega_1\). Again, \(\vec{\omega}\) will precess around the principal axis it started near to.

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

and \(\tilde{\omega}_3 = \sqrt{A}\omega_3\) and \(\tilde{\omega}_1 = \sqrt{C}\omega_1\). Notice the sign is the same in the last two terms. This no longer describes motion in a circle, but rather the dynamics will leave the region of \(\hat{x}_2\) exponentially quickly.

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

must hold throughout the evolution—but this is the equation of a sphere of radius \(L\) in the \((L_1,L_2,L_3)\) space. Incidentally, why does the direction of \(\vec{L}=(L_1,L_2,L_3)^T\) not need to be conserved also?

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

and using \(\vec{L} = (L_1, L_2, L_3)^T = (I_1 \omega_1, I_2 \omega_2, I_3 \omega_3)^T\), we arrive at which is an ellipsoid in \((L_1,L_2,L_3)\) space, with semi-axes \(\sqrt{2TI_1}\), \(\sqrt{2TI_2}\), and \(\sqrt{2TI_3}\).

Click figure to download the CDF demo.

Conservation of energy and momentum require that the angular momentum vector lie at the intersection of an ellipsoid with a sphere (shown in blue).

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.

Click figure to download the CDF demo.

Contours of total angular momentum showing the possible orbits of the angular momentum vector \(\vec{L}\).

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

We can see from the equation that if the object as re-orient itself for towards the axis with the highest moment of inertia, then the total energy can drop. In order to continue to conserve angular momentum, \(\vec{L}\) has to continue to lie on the sphere \eqref{eq:sphere}. Try adjusting the total energy \(T\) in the simulation, as \(T\) drops, \(\vec{L}\) will become increasingly constrained near the axis with the largest moment of inertia.

The effect can be summarised glibly as flying saucers make better spacecraft than flying cigars!