Physics — Rigid Body Dynamics

Rattleback

The rattleback (also called a Celtic stone or celt) is a semi-ellipsoidal top with a curious chiral asymmetry: it spins freely in one direction but quickly destabilizes and reverses when spun the other way. A pitching instability feeds angular momentum from spin into rocking, then back into spin — but in the opposite direction. No external asymmetry is needed; the geometry alone breaks time-reversal symmetry in the spin.

𝓜̇ = I⁻¹(τ − ω × Iω) Euler rigid-body equations with geometric coupling

Spin ωz 0.00 rad/s

Pitch 0.00 rad/s

Roll 0.00 rad/s

Reversals 0

Live

Initial Spin Speed 8 rad/s

Chirality Offset 15 deg

Damping 0.008

Aspect Ratio a/b 2.5

Click canvas to give a random perturbation. Spin CW to see the instability; CCW to see stable spin.

How the Rattleback Works

A rattleback is a solid of revolution that is not symmetric about its vertical axis of spin. The principal axes of inertia are skewed relative to the geometric axes of the shell. This small angular offset — the chirality angle α — is enough to couple the three modes of oscillation: spin (ωz), pitch (ωx), and roll (ωy).

When you spin the rattleback in its unstable direction, small perturbations excite pitching oscillations. The pitch mode is coupled back to the spin mode with the opposite sign — energy is extracted from the spin and pumped into pitch, then eventually returned as spin in the opposite direction. The mechanism is entirely Newtonian; there is no friction paradox or violation of conservation laws.

Stable direction: spin energy is not efficiently coupled to pitch; the stone spins freely until damped.
Unstable direction: coupling is strong; one full reversal cycle can take under a second at high chirality.
Multiple reversals: if energy loss is low enough, the stone can reverse 2–3 times before coming to rest.
Historical note: Celtic stone artifacts found in British museums show this effect; Walker described it in Scientific American (1979).

The Equations

The dynamics follow Euler's equations for rigid-body rotation on a curved surface. Let the rattleback have principal moments I₁, I₂, I₃ with inertia axes rotated by angle α from the geometric axes. The no-slip contact constraint on a flat surface adds reaction forces. The coupling terms proportional to sin(2α) drive the spin reversal — at α = 0 or 90° there is no coupling and no reversal.