Helia: Hénon–Heiles System
Mechanics, Chaos & Fluids arc, page 94 of 193. Back to The Pixelated Physics Universe
Who they are
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.
Helia is a character of the Mechanics, Chaos & Fluids arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
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.
$$ \begin{gathered} H = \tfrac{1}{2}\left(p_x^{2} + p_y^{2}\right) + \tfrac{1}{2}\left(x^{2} + y^{2}\right) + x^{2}y - \tfrac{1}{3}y^{3} \\ \text{saddles at } E = \tfrac{1}{6} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $H$ | total energy (scaled, dimensionless units) | 1 |
| $x,\ y$ | positions (scaled) | 1 |
| $p_x,\ p_y$ | momenta (scaled) | 1 |
| $E$ | energy of the orbit (scaled) | 1 |
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.
Sources
- Hénon & Heiles (1964) The applicability of the third integral of motion: Some numerical experiments, Astron. J. 69, 73
- Scholarpedia: Hénon–Heiles system
Crossovers
Helia meets Arnold's Cat (Cat Map Chaos). Arnold's Cat stretches and folds a square until it is scrambled, the model of strong chaos. Helia's galaxy is mixed: at low energy her orbits stay regular, and at high energy they turn chaotic.
Helia meets Aurea (KAM Theorem). Helia's regular orbits at low energy are Aurea's surviving tori. As energy rises, more of them break up, and Helia's chaotic sea grows.
Wear the lore
- Helia – The Hénon–Heiles System Physics T-Shirt
- Related infographic poster: The Lyapunov Time of the Planets – Plate 081 Physics Poster
Concept keywords: Hénon–Heiles System, Helia, Mechanics, Chaos & Fluids, physics lore, Pixelated Physics