Index波、混沌与物质Hoodie

Poincaré Section Hoodie

ISLANDS IN A CHAOTIC SEA | Pixelated Physics

\[H = \tfrac12\left(p_x^2+p_y^2\right) + \tfrac12\left(x^2+y^2\right) + x^2y - \tfrac13y^3\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

Henri Poincaré’s trick for seeing motion in four dimensions: don’t watch the whole orbit. Mark a dot each time it passes through one plane. A Poincaré section of the Hénon–Heiles system at E = 1/8: bright islands of regular orbits inside a scattered sea of chaotic ones. 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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Returns & replacements21 days

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The physics

The Hénon–Heiles potential is a simple model of a star moving in a galaxy. Recording each time an orbit crosses one plane gives a Poincaré section: regular orbits trace smooth closed curves (islands), while chaotic ones scatter points across the 'sea'.

Henri Poincaré’s trick for seeing motion in four dimensions: don’t watch the whole orbit. Mark a dot each time it passes through one plane.

Michel Hénon and Carl Heiles applied it in 1964 to a simple model of a star moving in a galaxy. Every orbit here has the same energy, E = 1/8. Each was integrated numerically and cut at x = 0 going in the same direction.

Regular orbits leave dots on closed curves, the bright nested islands. Chaotic orbits scatter their dots over a region, the dim sea. Both kinds exist side by side at one energy, and an orbit never crosses from one to the other.

Raise the energy and the sea floods most of the islands. At E = 1/8 the islands still hold.

Islands in a chaotic sea.

Equations

\[H = \tfrac12\left(p_x^2+p_y^2\right) + \tfrac12\left(x^2+y^2\right) + x^2y - \tfrac13y^3\]
\[E = \tfrac18,\qquad \text{section } x = 0,\; p_x > 0\]
SymbolMeaningUnit
\(H\) Hamiltonian, total energy (scaled) \(\mathrm{1}\)
\(x, y\) position coordinates \(\mathrm{1}\)
\(p_x, p_y\) momenta \(\mathrm{1}\)
\(E\) energy of every orbit shown \(\mathrm{1}\)

Sources

  1. Poincaré (1890) Sur le problème des trois corps et les équations de la dynamique, Acta Math. 13, 1
  2. Hénon & Heiles (1964) The applicability of the third integral of motion: Some numerical experiments, Astron. J. 69, 73
  3. Lichtenberg & Lieberman, Regular and Chaotic Dynamics, 2nd ed. (1992)

Source Hénon, M.; Heiles, C. (1964). The applicability of the third integral of motion: Some numerical experiments. The Astronomical Journal 69, 73.doi.org/10.1086/109234

Field notes

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