Aurea

Aurea carries a ring of gold. When the system is nudged, the rational rings around her shatter, and hers holds.

The Physics

Take an integrable Hamiltonian system, whose motion winds around nested invariant tori, and perturb it slightly. Kolmogorov announced, and Arnold and Moser proved, that most tori survive. Those whose frequency ratios are “sufficiently irrational” (a Diophantine condition) persist, slightly deformed. Tori with rational, resonant frequency ratios break into island chains and chaotic layers. For small perturbations the surviving set has large measure.

For one model, the Chirikov standard map, Greene found numerically in 1979 that the last surviving rotational torus has the golden mean as its rotation number, and breaks at K ≈ 0.971635.

|k · ω| ≥ γ / |k|^τ

• k: any nonzero integer vector
• ω: frequency vector of the torus
• γ, τ: positive constants of the Diophantine condition
• Golden mean: (1 + √5)/2 = 1.618…

Kolmogorov 1954; Arnold, Russ. Math. Surveys 18(5), 9 (1963); Moser 1962 (area-preserving twist maps); Greene, J. Math. Phys. 20, 1183 (1979).

Honest caveat: “The golden torus is last” is a numerical result for the standard map, not a general theorem. KAM does not prove the Solar System is stable.

On the shirt

Front: Aurea, the pixel-art character, with “THE TORI SURVIVE”. Back: the survival rule, the Diophantine condition, Greene's golden-torus value, the golden mean and the 1954–1979 sources. Printed on a unisex tee.