Spin it like a frisbee — watch what happens
Any object flying freely in space has exactly three principal axes of rotation — three distinct ways to spin it. Your phone is a perfect example: spin it like a baton (around the long axis), flat like a frisbee, or flip it end-over-end by the side edges.
Any rigid body possesses three mutually perpendicular principal axes of inertia — eigenvectors of the inertia tensor I, with scalar moments I₁ ≤ I₂ ≤ I₃. For a typical smartphone (≈ 75 × 150 × 8 mm): I₁ ≈ long axis (smallest, ~470 g·cm²), I₂ ≈ width/toss axis (intermediate, ~1880 g·cm²), I₃ ≈ face normal (largest, ~2340 g·cm²).
Spin your phone like a baton (around the long axis) or flat like a frisbee and the rotation is stable. Even if you nudge it off-axis, the wobble bounces back — like a ball in a bowl. The animation shows a real small wobble that stays bounded.
Euler's rotation equations predict rotation about I₁ and I₃ is Lyapunov-stable. Linearizing about ω = ω₀ê₁ gives δ̈ω ~ −λ²δω (λ² > 0) — a harmonic oscillator. The perturbation oscillates at frequency λ but never grows. Both extremal axes sit at energy minima/maxima on the angular-momentum sphere, so perturbations have a restoring force.
Try it: toss your phone so it flips end-over-end by rotating around the side edges. A tiny wobble off that axis doesn't bounce back — it amplifies itself. The tilt feeds spin into the long axis, which feeds into the face axis, which reverses the flip direction. Chain reaction, not gradual drift. The graph shows the signature: long flat plateaus near ±4 rad/s, then a sudden flip. Smaller wobble → same flips, just slower.
Near I₂, the Euler equations give δ̈ω ~ +λ²δω — a positive feedback (λ² = ω₀²(I₃−I₂)(I₂−I₁)/(I₁I₃) > 0). I₂ is a saddle point of kinetic energy T on the angular-momentum sphere — unlike I₁ (energy max) and I₃ (energy min), it has no restoring force. The ω-vector traces a heteroclinic orbit connecting the two I₂ poles, which is why the face-spin component ωz steps between ±|ω| rather than oscillating smoothly. Flip period: T = 2K(k)/λ.
On Earth, your hand dampens the tumbling almost instantly. In space there's nothing to slow it down. Notice the gold marker — that's the angular momentum vector L, which stays perfectly fixed in space while the object tumbles wildly around it. Soviet cosmonaut Vladimir Dzhanibekov discovered this with a wing nut aboard Salyut 7 in 1985. Your phone would do the same.
Euler's equations conserve both T = ½ΣIᵢωᵢ² and |L|² = ΣIᵢ²ωᵢ². The ω-vector must lie on the intersection of the energy ellipsoid and the angular-momentum sphere simultaneously. Near I₂ this intersection is a figure-8 heteroclinic orbit connecting two unstable fixed points — the trajectory crosses between hemispheres each flip. The gold indicator shows L fixed in the lab frame; the object tumbles around it indefinitely without dissipation.
Toss your phone end-over-end around the side edges — it tumbles. Throw a book — it tumbles. This isn't a quirk of tennis rackets; it happens to any rigid body with three distinct moments of inertia. The intermediate axis is a saddle point of kinetic energy on the angular-momentum sphere: stable when perturbed in one plane, unstable in the other. Asteroids tumble this way. Spacecraft engineers must account for it — satellites need active nutation dampers to bleed energy out of the unstable wobble mode and stay pointing at their targets.