M056 · Free rigid body / intermediate-axis instability

Tennis-Racket Theorem

Release an asymmetric rigid body about each principal axis. Orbit the three-dimensional body, seed a tiny transverse spin, and compare half-turn reversals with stable rotation. Follow body angular velocity along the intersection of its momentum sphere and 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 $\text{Pending}$

PREPARE / COMPARE / VERIFY

Prepare the rigid body

Show angular momentum and axesArrows have independently normalized lengths. Dimensions share one scale; Earth and orbit are inset 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 dimensions; their inertia order can change when you edit the shape. The two unselected initial angular velocities each equal the transverse fraction times the absolute principal spin. Drag the orange handle to change this fraction. Exact unperturbed rotation does not test stability.

Uniform rectangular solid in free space, with no external torque, damping, deformation or collisions. The name identifies the theorem; the modeled apparatus is a cuboid, not a calibrated tennis racket. Face stripes are massless visual markings. The camera follows its centre of mass; translation is decoupled. Principal-axis numbering follows dimensions and need not match inertia order. Linear growth or oscillation rates are local estimates; large flips use the full nonlinear equations. Exactly unperturbed principal rotation remains an exact solution, including the unstable axis. The projected sphere and ellipsoid share body-frame momentum coordinates. Full-record curves include future calculated samples. Fast spin may alias at screen frame rates; pause and use fine steps.

Start with the essentials

Focus question: Which principal axis amplifies a tiny transverse spin?
One-sentence intuition: The intermediate inertia gives exponential local growth. Nonlinear motion repeatedly reverses a body axis without changing the conserved world vector.

Core mathematical model

Principal inertia

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 reorder the inertias; numbering stays attached to the body.

Euler equations

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

No torque does not mean constant body angular velocity. The body frame itself rotates.

World angular momentum

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

The inertial vector is fixed while its components in the moving body frame 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 constraints use body angular momentum, not angular velocity. The teal trace lies on their intersection.

Local intermediate-axis growth

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

This positive growth rate applies when the prepared axis has strictly intermediate inertia. It is an infinitesimal estimate, not a flip frequency.

Stable-axis oscillation

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

The product is positive for the smallest and largest inertias. A repeated inertia removes the exponential prediction; it does not imply dissipative settling.

Common difficulties

No torque means no flip

Typical misconceptionAngular velocity must stay fixed if nothing pushes the object.

Better mental modelWorld angular momentum is fixed; an anisotropic rotating inertia tensor changes angular velocity.

The second axis is always intermediate

Typical misconceptionThe number of an axis determines its stability.

Better mental modelRead the inertias after editing dimensions; numbering remains body-fixed.

Run the experiment

01
Compare the axes
Use minimum, intermediate and maximum presets, then review the full record.
What to observe: The intermediate release reverses its selected axis; the other two keep small deviations.
02
Perturb the exact solution
Compare the exact and tiny-perturbation presets; drag the orange amplitude handle.
What to observe: An unstable solution stays exact without a perturbation; a smaller seed delays departure.
03
Follow the intersection
Pause near a flip and inspect body angular velocity and the momentum geometry.
What to observe: The body components move along both conserved surfaces while the world momentum stays fixed.
04
Change geometry
Make two or three dimensions equal, then compare mass scaling and step-size defects.
What to observe: Repeated inertias change the local growth law; mass rescales energy and momentum without changing attitude.