Hénon–Heiles · Poincaré section

How does a smooth Hamiltonian system slide from order into chaos as its energy rises?

▶ Run the simulationSee the measured result

How the lab tests it

Integrate orbits of the Hénon–Heiles galactic potential and take a Poincaré section (their crossings of the plane x=0); a regular orbit pierces it on a smooth closed curve, a chaotic one scatters. Each orbit's Lyapunov exponent colours it blue (regular) or warm (chaotic).

What it looks for

the KAM scenario: at low energy nested invariant tori (regular), which progressively break into a chaotic sea as the energy climbs toward the escape energy 1/6

Hénon–Heiles escape energy calculator — the barrier found rather than quoted: saddle channels, the exact triangular equipotential, and the leapfrog's own bias

Hénon and Heiles asked in 1964 whether a star in a galaxy has a third conserved quantity, and their answer turned on one number: orbits in V = ½(x²+y²) + λ(x²y − y³/3) stay bound only below the critical energy 1/6. This page does not quote that number. It declares the potential and nothing else, then goes looking for the barrier — three Newton searches on ∇V = 0 launched from three generic points whose only scale is the problem's own length 1/λ — and reads the escape energy off the potential at whatever points the search lands on. What comes back is the closed form to the last digit, and with it three things the finding states but never shows: that all three channels sit at exactly the same radius, that the barrier's curvature is the SAME at every coupling while its height is not, and that the escape equipotential is not a curve at all but three exact straight lines, which this page tests by evaluating V along them in your browser rather than asserting. The λ-scaling exponent is measured too, as a free two-point slope between independent saddle hunts, so the −2 is an output. The last direction turns the same machinery on the simulation's own integrator: the finding discloses that the shipped leapfrog reports the barrier's instability 2.604e-5 low, and that bias has an exact closed form whose first two series terms account for the disclosed figure and its residue completely. Four things this page will NOT do. It will not tell you what an orbit DOES: whether a given launch is regular or chaotic is a question about trajectories that the simulation answers by integrating them for 160 section crossings, and no formula here replaces that. It will not give an escape TIME — above the barrier the channels are open, but how long a particular orbit takes to find one diverges as the energy comes down to E_esc, and that is something the lab measures by waiting. It will not price the KAM transition: the energies at which invariant curves break have no closed form, and the three chaotic fractions below are the lab's readings placed in the barrier's own units, not predictions. And it carries no physical units at all — this is the 1964 model in its own scaled variables, and a real galactic potential needs a fit that nothing here tests.