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Double Pendulum Hoodie

TWO HINGES ARE ENOUGH | Pixelated Physics

\[L = ml^2\left[\dot\theta_1^2 + \tfrac12\dot\theta_2^2 + \dot\theta_1\dot\theta_2\cos(\theta_1-\theta_2)\right] + mgl\left(2\cos\theta_1 + \cos\theta_2\right)\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

Two rigid rods, two hinges, gravity. Nothing else. The path, computed step by step, of a double pendulum released at rest from 150°, 150°: two hinges, no friction, no repeat. Chaos from Newton alone. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

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Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

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Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

Two pendulums hinged together follow exactly from one Lagrangian with no randomness at all. Yet for large swings the motion is chaotic: two starts that differ by a hair soon follow completely different paths.

Two rigid rods, two hinges, gravity. Nothing else.

The equations of motion come straight from the Lagrangian on the back. There is no noise and no randomness. Released from rest with both rods at 150°, the lower bob traces the path in the art over 18 seconds, coloured by time, and the rods are drawn where they end up.

Start it again with an angle changed in the sixth decimal place and the two runs agree for a while, then part completely. The system is chaotic: errors grow exponentially, so prediction has a horizon even though the rules are exact.

Shinbrot and colleagues tested this on a real double pendulum in 1992: tiny differences in the release grew until the runs no longer matched.

Two hinges are enough.

Equations

\[L = ml^2\left[\dot\theta_1^2 + \tfrac12\dot\theta_2^2 + \dot\theta_1\dot\theta_2\cos(\theta_1-\theta_2)\right] + mgl\left(2\cos\theta_1 + \cos\theta_2\right)\]
\[\frac{d}{dt}\frac{\partial L}{\partial\dot\theta_i} = \frac{\partial L}{\partial\theta_i}\]
SymbolMeaningUnit
\(L\) Lagrangian, kinetic minus potential energy \(\mathrm{J}\)
\(m\) mass of each bob \(\mathrm{kg}\)
\(l\) length of each rod \(\mathrm{m}\)
\(g\) gravitational acceleration \(\mathrm{m/s^2}\)
\(\theta_1, \theta_2\) angles of the upper and lower rods from vertical \(\mathrm{rad}\)
\(\dot\theta_1, \dot\theta_2\) angular velocities \(\mathrm{rad/s}\)

Sources

  1. Lagrange (1788) Mécanique analytique, Paris
  2. Shinbrot, Grebogi, Wisdom & Yorke (1992) Chaos in a double pendulum, Am. J. Phys. 60, 491
  3. Taylor, Classical Mechanics (2005), Ch. 7 and Ch. 12

Source Lagrange, J.-L. (1788). Mécanique analytique. Paris: Veuve Desaint.

Field notes

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