Double Pendulum — Poincaré Section
Hamiltonian chaos: trajectory (left) and Poincaré map at θ₂=0 plane (right)
Points: 0
Energy: -
Lyap est: -
t = 0
The double pendulum (unit masses, unit lengths, g=1) is a canonical Hamiltonian system exhibiting
chaos for most initial conditions at moderate energy. The Hamiltonian
H = (p₁²+2p₂²-2p₁p₂cosΔ)/(2(2-cos²Δ)) - 2cosθ₁ - cosθ₂ is conserved.
The Poincaré section (right) records (θ₁, p₁) whenever θ₂ = 0 and ṗ₂ > 0.
Regular orbits produce closed curves (KAM tori); chaotic trajectories fill 2D regions.
Integrable separable cases appear as distinct islands of stability surrounded by a chaotic sea.