Hamiltonian Chaos

Compare regular and irregular motion

Choose a model and initial state. Choices update immediately; physical inputs after a short pause or Enter. Each prepares a paused experiment at zero. Play and Step advance it. Phase portrait and 3D orbit share the same state; switching between them pauses without restarting. Entering or leaving the grid starts a new experiment. Speed and trace length change the current experiment without resetting its time. The nearby start shifts x (θ₁ on the torus) by ε, which can also change Hénon–Heiles energy. It is not a same-energy Lyapunov measurement. Compare lower and higher E₀ at λ=1, or K=0 with K=5. A finite trace alone does not prove chaos.

Equations and coordinates

Hénon–Heiles: H=(pₓ²+pᵧ²+x²+y²)/2+λ(x²y−y³/3). Initially x=0,pᵧ=0 and pₓ is positive and chosen from E₀,y₀. Velocity Verlet uses a selectable Δt (default 0.02). Energy drift is numerical, not physical dissipation; reduce the time step in Playback and mesh for quantitative convergence studies. For λ=1, escape channels open at E=1/6; other couplings change this threshold. Orbits can escape, including a perturbed one.

Standard map on the cylinder: p′=p+K sin(x), x′=(x+p′) mod 2π; momentum is not wrapped. Torus rotation: θ₁′=(θ₁+α) mod1, θ₂′=(θ₂+β) mod1. It is regular, with constant short-arc separation, and provides a comparison. Maps are plotted as discrete points; joining jumps would invent continuous motion.

Mesh and playback

The mesh uses shared sampled vertices and straight edges. Standard-map angles and torus angles are unwrapped here. The standard map and torus translation have determinant 1, but finite straight-edge polygons need not retain their area after nonlinear deformation. Hénon–Heiles is a four-dimensional symplectic flow; its (x,pₓ) projection does not have to preserve area. This view therefore does not claim a measured Liouville ratio.

Playback targets one model time unit/s or 20 map iterations/s, multiplied by the rate. Fixed steps and a fractional accumulator prevent frame-rate dependent model time. Under heavy load the animation can slow. One step always performs one integration step or map iteration. The retained trace is bounded; no simulation runs before Run. Hidden tools suspend playback. PNG pauses and captures the current view and parameters. 3D is available for Hénon–Heiles with WebGL; rotate by dragging and zoom with Ctrl/Meta-wheel.

Chirikov & Shepelyansky · Meiss · Hénon–Heiles

Frame orbit fits the retained 3D paths. Colored endpoint markers show the current states. Camera position, target and clipping range are kept with JSON and reload. Plain wheel scrolls the page; Ctrl/Meta-wheel zooms the 3D camera.

Hénon–Heiles reconstructs at most the retained 4000-point Verlet window from its exact oldest state; a long experiment is never replayed from its beginning. Discrete maps retain their actual samples, so chaotic sin evaluations are not repeated during restore. Grid vertices are stored directly. Loading never starts playback.

Only input values are remembered in this browser. Calculated results and experiment histories are not saved.