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Pendulum Phase Space Hoodie

SWING INSIDE, SPIN OUTSIDE | Pixelated Physics

\[H = \frac{p^2}{2ml^2} - mgl\cos\theta\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

Can every possible motion of a pendulum fit on one picture? This phase portrait plots angle against angular momentum. Because energy is conserved, each motion traces one curve: low energies make closed loops, where the pendulum swings, and high energies make open curves, where it spins over the top. The separatrix divides the two. This is a black pixel-art hoodie from Pixelated Physics.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

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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Shipping calculated at checkout by destination.

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

In a pendulum's phase space, small swings trace closed loops, while fast motions go all the way over the top and keep spinning. The boundary between the two, the separatrix, is the motion with exactly enough energy to reach the top and stop there.

Every possible motion of a pendulum fits on one map.

Across the map runs the angle θ, up the map the angular momentum p. Energy is conserved, so each motion follows one curve of constant H = p²/2ml² − mgl cos θ, the Hamiltonian.

Low energies close into eyes around the hanging-down position: the pendulum swings back and forth. High energies run along the top and bottom without closing: the pendulum goes over the top and spins. One curve, the separatrix at E = mgl, divides them. It passes through the upside-down balance point, where p = ±2ml²ω cos(θ/2) reaches zero.

The art repeats every 2π, because θ and θ + 2π are the same position. The cream dots are stable rest points; the amber dots are the unstable balance points.

Swing inside, spin outside.

Equations

\[H = \frac{p^2}{2ml^2} - mgl\cos\theta\]
\[E = mgl \;\text{(separatrix)}\]
\[p = \pm 2ml^2\omega\cos\frac{\theta}{2},\qquad \omega = \sqrt{g/l}\]
SymbolMeaningUnit
\(H\) energy, the Hamiltonian \(\mathrm{J}\)
\(p\) angular momentum about the pivot \(\mathrm{kg·m^2/s}\)
\(\theta\) angle from the bottom \(\mathrm{rad}\)
\(m\) bob mass \(\mathrm{kg}\)
\(l\) length of the rod \(\mathrm{m}\)
\(g\) gravitational acceleration \(\mathrm{m/s^2}\)
\(E\) energy of one curve \(\mathrm{J}\)
\(\omega\) small-swing angular frequency \(\mathrm{rad/s}\)

Sources

  1. Hamilton (1834) On a General Method in Dynamics, Phil. Trans. R. Soc. 124, 247
  2. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Sec. 6.7: Pendulum
  3. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., Ch. 8

Source Hamilton, W. R. (1834). On a general method in dynamics; by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function. Philosophical Transactions of the Royal Society of London, 247–308.doi.org/10.1098/rstl.1834.0017

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