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Tennis Racket Theorem

Spin a phone, a book, or a tennis racket about its three axes and one of them misbehaves. A free rigid body has three principal axes, and Euler's equations make spin about the largest and smallest moments of inertia stable, but spin about the middle one unstable: the slightest wobble grows until the body suddenly flips end over end, then flips back, over and over. This is the tennis-racket theorem, the same Dzhanibekov effect a cosmonaut filmed a wing-nut doing in orbit. Nothing pushes it; angular momentum and energy are both exactly conserved through every flip. The scene tumbles the body under Euler's equations; the diagnostic plots the three body-frame spin rates, where the middle one periodically reverses sign, each reversal a flip.

Figure 1. Free rigid-body rotation and the intermediate-axis instability. Top: the body tumbling under Euler's equations, with its three principal axes; spun about the middle axis it periodically flips. Bottom: the body-frame spin rates ω₁, ω₂, ω₃ over time, the middle one reversing sign at each flip. Method: RK4 on Euler's equations with quaternion orientation.
objectT-handle
spin axismiddle
spin rate6
nudge0.040

WHAT TO TRY

  • Keep the spin axis on middle and watch the body flip end over end, again and again, with no force applied.
  • Switch to the major or minor axis: now it spins smoothly and never flips, those axes are stable.
  • On the middle axis, shrink the nudge: the flips just take longer to begin; any nonzero wobble eventually triggers them.
  • Watch the diagnostic: between flips the middle spin rate holds steady, then snaps to the opposite sign as the body turns over.
  • Switch the object and watch the principal-moment bars (top right) and the $I_1:I_2:I_3$ readout change: each shape's moments are computed from its real 3D dimensions, so the thin phone, the chunky book, the racket and the T-handle each get a different inertia and a different measured flip period, even though all four flip about their middle axis. Same equations, genuinely different numbers, not one case in different skins.