Helia

Helia 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.

The Physics

In 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.

At 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.

H = ½(p_x² + p_y²) + ½(x² + y²) + x²y − y³/3

• H: total energy (dimensionless units)
• x, y: positions
• p_x, p_y: momenta
• Saddle energy: E = 1/6

Hénon & Heiles, Astronomical Journal 69, 73 (1964). It became a textbook early example of chaos in a Hamiltonian system.

Honest caveat: It was not the first chaos ever found: Poincaré came first. Whether a real galaxy behaves like this toy model is a modelling question.

On the shirt

Front: 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.