M056 · Intermediate-axis instability of a free rigid body

Tennis-Racket Theorem

Why does a free object spinning about its middle axis keep flipping over? Because spin about the middle axis is unstable. A tiny wobble about the other axes grows exponentially at first, and the body flips over, again and again, while its angular momentum and energy stay exactly the same. Spin about the axis with the largest or smallest moment of inertia only wobbles a little.

Spin an asymmetric rigid body about each principal axis and add a tiny sideways spin. Turn the three-dimensional view and compare the half-turn flips with steady rotation. Follow the body angular velocity and the curve where the angular-momentum sphere meets the energy ellipsoid.

Physical elapsed time $\text{Pending}$

First principal inertia $\text{Pending}$

Second principal inertia $\text{Pending}$

Third principal inertia $\text{Pending}$

First body angular velocity $\text{Pending}$

Second body angular velocity $\text{Pending}$

Third body angular velocity $\text{Pending}$

Selected-axis deviation $\text{Pending}$

Body rotational energy $\text{Pending}$

Total angular momentum $\text{Pending}$

Prepared state $\text{Pending}$

Local growth / libration rate $\text{Pending}$

Max scaled balance defect $\text{Pending}$

Two-step attitude difference $\text{Pending}$

World momentum defect $\text{Pending}$

Set up, compare, then verify

Prepare the rigid body

Show angular momentum and axesEach kind of arrow is normalized on its own. The body dimensions share one scale; Earth and the orbit appear only as an inset for context.Body mass

m

1.00\,\mathrm{kg}

First-axis dimension

a

0.60\,\mathrm{m}

Second-axis dimension

b

0.35\,\mathrm{m}

Third-axis dimension

c

0.08\,\mathrm{m}

Prepared principal axis

j

2.00\,

Initial principal-axis spin

\Omega

8.00\,\frac{\mathrm{rad}}{\mathrm{s}}

Transverse spin fraction

\epsilon

0.02\,

Record duration

T

8.00\,\mathrm{s}

Reviewed fraction

t/T

0.00\,\%

Axes 1, 2 and 3 follow the body's dimensions, so their order by moment of inertia can change when you edit the shape. The initial angular velocity about each of the other two axes equals the transverse fraction times the magnitude of the spin about the chosen axis. Drag the orange handle to change this fraction. A perfectly unperturbed rotation cannot test stability.

The body is a uniform rectangular block in free space, with no external torque, damping, deformation or collisions. The theorem is named after a tennis racket, but the Lab models a block, not a calibrated racket. The stripes on its faces are massless markings. The camera follows the centre of mass, and translation is decoupled from rotation. Axes are numbered by the block dimensions, so the numbers need not follow the order of the moments of inertia. The linear growth and oscillation rates are local estimates; large flips are computed from the full nonlinear equations. Spin exactly about a principal axis with no perturbation remains an exact solution, even about the unstable axis. The sphere and ellipsoid are drawn in the same body-frame angular-momentum coordinates. Curves for the full record include samples already calculated for later times. Fast spin can alias at the screen frame rate; pause and use fine steps.

Start with the essentials

Focus question: About which principal axis does a tiny sideways spin grow?
One-sentence intuition: About the axis of intermediate inertia, a small wobble first grows exponentially; the nonlinear motion then flips a body axis over again and again, while the conserved angular momentum vector stays fixed in space.

Core mathematical model

Principal moments of inertia of a block

I_1=\frac{m(b^2+c^2)}{12},\quad I_2=\frac{m(a^2+c^2)}{12},\quad I_3=\frac{m(a^2+b^2)}{12}

Changing a dimension can change the order of the moments of inertia, but the axis numbers stay attached to the body.

Euler equations

I_1\dot\omega_1=(I_2-I_3)\omega_2\omega_3\quad\text{and cyclic permutations}

Even with no torque, the angular velocity measured in the body frame can change, because the body frame itself rotates.

Angular momentum in the fixed frame

\mathbf L=R I\boldsymbol\omega_b,\qquad\dot{\mathbf L}=\mathbf 0

The vector stays fixed in the inertial frame, while its components along the moving body axes change.

Momentum sphere and energy ellipsoid

\sum_i L_{b,i}^2=|\mathbf L|^2,\qquad\sum_i\frac{L_{b,i}^2}{I_i}=2E

Both surfaces are written in terms of the body-frame angular momentum, not the angular velocity. The teal trace lies on the curve where they meet.

Growth rate near the intermediate axis

\lambda=|\Omega|\sqrt{\frac{(I_j-I_{\min})(I_{\max}-I_j)}{I_{\min}I_{\max}}}

This positive growth rate applies when the chosen axis has a moment of inertia strictly between the other two. It describes how an infinitesimal wobble starts to grow; it is not the flip frequency.

Oscillation about a stable axis

\nu=|\Omega|\sqrt{\frac{(I_j-I_k)(I_j-I_\ell)}{I_k I_\ell}}

The product under the square root is positive for the smallest and largest inertias. When two inertias are equal, the exponential prediction no longer applies, but that does not mean the motion settles down by losing energy.

Common difficulties

A flip with no torque

Typical misconceptionIf nothing twists the object, its angular velocity should stay the same.

Better mental modelWhat stays fixed is the angular momentum in the fixed frame. The body has different moments of inertia along different axes, and as it turns, the same angular momentum corresponds to a changing angular velocity.

Axis numbers do not decide stability

Typical misconceptionThe second axis is always the intermediate one, so its number tells you whether it is stable.

Better mental modelThe numbers stay attached to the body. After you change the dimensions, read the three inertias to see which axis is in the middle.

Run the experiment

01
Compare the axes
Run the Minimum-inertia axis, Intermediate-axis flip and Maximum-inertia axis presets, then use Review full record.
What to observe: Spun about the intermediate axis, the body flips that axis over. About the other two axes, the deviation stays small.
02
Perturb the exact solution
Compare the Exactly unperturbed and Tiny perturbation presets, then drag the orange handle to change the perturbation size.
What to observe: With no perturbation, even the unstable spin stays exact. The smaller the initial perturbation, the later the body departs from it.
03
Follow the intersection curve
Pause near a flip and look at the body angular velocity and the angular-momentum sphere and ellipsoid.
What to observe: The body-frame components move along both conserved surfaces at once, while the angular momentum in the fixed frame stays put.
04
Change the shape
Make two or three dimensions equal, then compare what changing the mass does and how the defects depend on step size.
What to observe: Equal inertias change how small deviations grow. Changing the mass rescales the energy and angular momentum but leaves the attitude unchanged.