{"product_id":"aurea-the-kam-theorem-physics-t-shirt","title":"Aurea – The KAM Theorem Physics T-Shirt","description":"\u003cp\u003eAurea carries a ring of gold. When the system is nudged, the rational rings around her shatter, and hers holds.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eTake 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.\u003c\/p\u003e\u003cp\u003eFor 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e|k · ω| ≥ γ \/ |k|^τ\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003ek: any nonzero integer vector\u003c\/li\u003e\n\u003cli\u003eω: frequency vector of the torus\u003c\/li\u003e\n\u003cli\u003eγ, τ: positive constants of the Diophantine condition\u003c\/li\u003e\n\u003cli\u003eGolden mean: (1 + √5)\/2 = 1.618…\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eKolmogorov 1954; Arnold, Russ. Math. Surveys 18(5), 9 (1963); Moser 1962 (area-preserving twist maps); Greene, J. Math. Phys. 20, 1183 (1979).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e “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.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: 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.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"S \/ White","offer_id":50960507994215,"sku":"22148626024454291616","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":50960508026983,"sku":"79039476803849779451","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":50960508059751,"sku":"23548641236099570103","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":50960508092519,"sku":"16131246341459832207","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":50960508125287,"sku":"68774887777928044277","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":50960508158055,"sku":"44364499396950921955","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":50960508190823,"sku":"28227015562455914905","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":50960508223591,"sku":"25379380442938054584","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":50960508256359,"sku":"33259140245631750839","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":50960508289127,"sku":"13939527985175482635","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":50960508321895,"sku":"52312085557849395568","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":50960508354663,"sku":"15806396625518047185","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":50960508387431,"sku":"30356304244187905281","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":50960508420199,"sku":"98728787670730047304","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":50960508452967,"sku":"30126114704340471049","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":50960508485735,"sku":"18562530372293387385","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/kam-theorem_front_16_white.png?v=1793625572","url":"https:\/\/pixelatedphysics.com\/products\/aurea-the-kam-theorem-physics-t-shirt","provider":"Pixelated Physics","version":"1.0","type":"link"}