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Torque-Free Rigid Body (Euler's Equations) 3D

Throw a rigid body with no torque on it and it does not keep spinning about a fixed line. Its instantaneous spin axis, the white arrow, drifts around inside the body and around in space, tracing a closed curve called the polhode. What stays nailed in place is the angular momentum, the gold arrow: with no torque it cannot change, so the whole tumble organises itself around that one fixed direction. Two numbers, the rotational energy and the magnitude of angular momentum, are exactly conserved, and together they pin the spin axis to a single curve on the inertia ellipsoid. Spin near the long or short axis and that curve is a tight loop, a steady tumble; spin near the middle axis and it runs along the separatrix, so the body flips end over end. The scene shows the ellipsoid tumbling with the spin axis and the fixed angular momentum; the diagnostic is the polhode itself, the path the spin axis traces.

Figure 1. Torque-free rigid-body rotation (Poinsot construction). Top: the inertia ellipsoid tumbling under Euler's equations, with the spin axis omega (white), the conserved angular momentum L (gold, fixed in space), and the polhode painted on the body. Bottom: the polhode traced in the omega1-omega3 principal plane, a tight loop for stable spins and a separatrix bowtie for the intermediate axis. Method: RK4 on Euler's equations with quaternion orientation.
spin axismiddle
asymmetry1.00
spin rate4.0
nudge0.05

WHAT TO TRY

  • Keep the spin axis on middle: the polhode below traces a bowtie through the centre and the ellipsoid flips end over end, on a clock.
  • Switch to the minor or major axis: the polhode collapses to a small closed loop and the tumble is steady, no flips.
  • Watch the gold angular-momentum arrow: it barely moves while everything else tumbles, because no torque acts on the body.
  • Raise the asymmetry: a more elongated ellipsoid makes the middle-axis flips faster and more violent.