Presets
Playback
Masses, lengths and gravity
Start state
Twin trajectory
Model checks
The science
Lagrangian equations
For angles θ1, θ2 from the vertical, the Lagrangian is L = T − U with T = ½(m1+m2)L1²ω1² + ½m2L2²ω2² + m2L1L2ω1ω2cos(θ1−θ2) and U = −(m1+m2)gL1cosθ1 − m2gL2cosθ2. The Euler-Lagrange equations are integrated with the same fourth-order Runge-Kutta scheme as the 3D page, at 2 ms steps. The total energy T + U is conserved.
Chaos
Two trajectories started a tiny angle ε apart separate roughly as ε eλt with a positive Lyapunov exponent λ for large angles, but stay close in the small-angle regime where the motion is a sum of two normal modes with (ad − bc)ω&sup4; − g(a + (m1+m2)d)ω² + (m1+m2)g² = 0, a = (m1+m2)L1, b = m2L2, c = L1, d = L2.
Honest simplifications: frictionless pivots, massless rods, point masses, no air drag. The separation curve saturates once the twins are completely decorrelated, so the exponent is read from the early rise only.
What to try
- Start the Small oscillations preset: both pendulums stay together.
- Choose Wild flip and watch the red twin leave the green one within seconds.
- Fast mode and Slow mode excite one normal mode each: perfectly periodic motion.