{"product_id":"bertrande-bertrands-theorem-physics-t-shirt","title":"Bertrande – Bertrand's Theorem Physics T-Shirt","description":"\u003cp\u003eBertrande traces two orbits at once: a Kepler ellipse in one hand, an oscillator ellipse in the other. Every other force law frays into a rosette behind her.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eAmong all central forces, only two make every bounded orbit close on itself, for any starting conditions: the inverse-square law and the linear (Hooke's-law) force. For inverse-square attraction the centre of force sits at a focus of the ellipse; for the isotropic oscillator it sits at the centre. Any other law lets the orbits precess into rosettes. A quick test uses near-circular orbits: β, the ratio of radial to orbital angular frequency, must come out as 1 or 2.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF = −k\/r²  or  F = −k r;   β² = 3 + r f′(r)\/f(r) = 1 or 4\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eF: radial force on the body (N), minus = attractive\u003c\/li\u003e\n\u003cli\u003er: distance from the force centre (m)\u003c\/li\u003e\n\u003cli\u003ek: force constant (Kepler N m², oscillator N\/m)\u003c\/li\u003e\n\u003cli\u003ef(r): magnitude of the attractive central force (N)\u003c\/li\u003e\n\u003cli\u003eβ: radial-to-orbital frequency ratio for near-circular orbits (dimensionless)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eJoseph Bertrand proved it in the Comptes Rendus in 1873 (vol. 77, p. 849). For a power law f ∝ rⁿ the test gives β² = n + 3, so n = −2 or n = 1.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The theorem is about ALL bounded orbits. Other force laws do allow some closed orbits; only these two force every orbit to close.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Bertrande, the pixel-art character, with the title. Back: the two closed-orbit laws, the β test, and three orbit panels (focus, centre, open rosette). Printed on a unisex tee.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"S \/ White","offer_id":50960823615591,"sku":"55483548943870005679","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":50960823648359,"sku":"29036256725150581190","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":50960823681127,"sku":"15017839554198044352","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":50960823713895,"sku":"12498137053996753370","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":50960823746663,"sku":"18380721396561677877","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":50960823779431,"sku":"30883153858855665724","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":50960823812199,"sku":"20572118056605727094","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":50960823844967,"sku":"15102688495434083114","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":50960823877735,"sku":"44342749274156709936","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":50960823910503,"sku":"91269272716559824214","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":50960823943271,"sku":"86828224982821860550","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":50960823976039,"sku":"13581442355511369690","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":50960824008807,"sku":"95813851933967458053","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":50960824041575,"sku":"16206960398653094911","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":50960824074343,"sku":"20253415229534452173","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":50960824107111,"sku":"26284166546511557123","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/bertrand-theorem_front_16_white.png?v=1793628515","url":"https:\/\/pixelatedphysics.com\/sv\/products\/bertrande-bertrands-theorem-physics-t-shirt","provider":"Pixelated Physics","version":"1.0","type":"link"}