Aurea: KAM Theorem
Mechanics, Chaos & Fluids arc, page 95 of 193. Back to The Pixelated Physics Universe
Who they are
Aurea carries a ring of gold. When the system is nudged, the rational rings around her shatter, and hers holds.
Aurea is a character of the Mechanics, Chaos & Fluids arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
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.
$$ \begin{gathered} |\mathbf{k}\cdot\boldsymbol{\omega}| \ge \frac{\gamma}{|\mathbf{k}|^{\tau}} \\ \text{Standard map: golden torus breaks at } K = 0.971635 \\ \frac{1+\sqrt{5}}{2} = 1.618\ldots \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $\mathbf{k}$ | any nonzero integer vector | 1 |
| $\boldsymbol{\omega}$ | frequency vector of the torus | rad/s |
| $\gamma$ | positive constant of the Diophantine condition | rad/s |
| $\tau$ | positive exponent of the Diophantine condition | 1 |
| $K$ | standard-map kick strength | 1 |
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.
Sources
- Arnold (1963) Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russ. Math. Surveys 18(5), 9
- Greene (1979) A method for determining a stochastic transition, J. Math. Phys. 20, 1183
- Scholarpedia: Kolmogorov-Arnold-Moser theory
Crossovers
Aurea meets Bertrande (Bertrand's Theorem). Bertrande says that only the inverse-square and linear forces close every bounded orbit. Aurea says that when such a system is slightly perturbed, most of its regular motion survives on invariant tori. Together they explain why the Solar System is so orderly.
Aurea meets Helia (Hénon–Heiles System). Helia's regular orbits at low energy are Aurea's surviving tori. As energy rises, more of them break up, and Helia's chaotic sea grows.
Wear the lore
Concept keywords: KAM Theorem, Aurea, Mechanics, Chaos & Fluids, physics lore, Pixelated Physics