Double pendulum - 2D chaos lab

Equations

For point masses m₁, m₂ on rigid massless rods L₁, L₂ the Lagrangian is L = T - V with T = ½(m₁+m₂)L₁²ω₁² + ½m₂L₂²ω₂² + m₂L₁L₂ω₁ω₂cos(θ₁-θ₂) and V = -(m₁+m₂)gL₁cosθ₁ - m₂gL₂cosθ₂. The two Euler-Lagrange equations are integrated with fourth-order Runge-Kutta (the same maths as the 3D page).

Chaos

Nearby trajectories separate as |δ(t)| ~ |δ₀|e^λt; the largest Lyapunov exponent λ is the slope of ln|δ| in the straight part of the middle graph, before the gap saturates at the size of the attractor. Small swings are regular, large ones are chaotic.

Honest simplifications: ideal point masses; friction is a simple linear damping on each angle, not a bearing model; the fitted exponent is an estimate from one finite run, not a converged spectrum.

• Compare the two twin offsets (1e-2 and 1e-6 degrees): when does the orange path leave the blue one?
• Small-swing preset: the phase portrait is a closed loop and the gap grows only slowly.
• Add friction and watch the energy graph fall.