{"product_id":"helia-the-henon-heiles-system-physics-t-shirt","title":"Helia – The Hénon–Heiles System Physics T-Shirt","description":"\u003cp\u003eHelia wanders a triangular well at the heart of a galaxy. At low energy her path is neat; push her harder and her trail frays into dust.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eIn 1964 Michel Hénon and Carl Heiles asked whether a star moving in an axisymmetric galaxy has a third conserved quantity besides energy and angular momentum. They reduced the question to a two-degree-of-freedom model and integrated orbits numerically.\u003c\/p\u003e\u003cp\u003eAt low energy the Poincaré section shows smooth closed curves, as if a third integral exists. As the energy rises, the curves break up and a chaotic sea spreads between surviving islands. The potential has three-fold symmetry, and its three saddle points sit at energy 1\/6; above that, orbits can escape.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eH = ½(p_x² + p_y²) + ½(x² + y²) + x²y − y³\/3\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eH: total energy (dimensionless units)\u003c\/li\u003e\n\u003cli\u003ex, y: positions\u003c\/li\u003e\n\u003cli\u003ep_x, p_y: momenta\u003c\/li\u003e\n\u003cli\u003eSaddle energy: E = 1\/6\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eHénon \u0026amp; Heiles, Astronomical Journal 69, 73 (1964). It became a textbook early example of chaos in a Hamiltonian system.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e It was not the first chaos ever found: Poincaré came first. Whether a real galaxy behaves like this toy model is a modelling question.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Helia, the pixel-art character, with “CHAOS IN THE GALAXY”. Back: the Hamiltonian, the three saddles at E = 1\/6, the smooth-curves-to-chaotic-sea note and the 1964 paper. Printed on a unisex tee.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"S \/ White","offer_id":50960506060903,"sku":"23460767927740158243","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":50960506093671,"sku":"32424091448946230008","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":50960506126439,"sku":"82095281106911757472","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":50960506159207,"sku":"27992050886876021108","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":50960506191975,"sku":"21538510903002203369","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":50960506224743,"sku":"40025941828474387946","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":50960506257511,"sku":"11995754538915309202","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":50960506290279,"sku":"10121757533136203738","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":50960506323047,"sku":"25944606331930855695","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":50960506355815,"sku":"43329119461479949435","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":50960506388583,"sku":"17951676197075752052","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":50960506421351,"sku":"31258871344109439555","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":50960506454119,"sku":"29203557767710533931","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":50960506486887,"sku":"48886682680504567708","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":50960506519655,"sku":"15569979693095008251","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":50960506552423,"sku":"19745469620054192668","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/henon-heiles_front_02_white_c614b7af-cde3-4c5f-ac88-a185481324fa.png?v=1793625545","url":"https:\/\/pixelatedphysics.com\/products\/helia-the-henon-heiles-system-physics-t-shirt","provider":"Pixelated Physics","version":"1.0","type":"link"}