Lagrangian Mechanics
The double pendulum is a classic example of a chaotic dynamical system. Small changes in initial conditions lead to dramatically different trajectories over time.
Equations of Motion
The system is described using Lagrangian mechanics, where L = T - V (kinetic minus potential energy).
For a double pendulum with masses m₁, m₂ and lengths L₁, L₂:
θ₁'' = [−g(2m₁+m₂)sin(θ₁) − m₂g·sin(θ₁−2θ₂) − 2sin(θ₁−θ₂)m₂(θ₂'²L₂ + θ₁'²L₁cos(θ₁−θ₂))] / [L₁(2m₁ + m₂ − m₂cos(2θ₁−2θ₂))]
θ₂'' = [2sin(θ₁−θ₂)(θ₁'²L₁(m₁+m₂) + g(m₁+m₂)cos(θ₁) + θ₂'²L₂m₂cos(θ₁−θ₂))] / [L₂(2m₁ + m₂ − m₂cos(2θ₁−2θ₂))]Numerical Integration
This simulation uses 4th-order Runge-Kutta (RK4) integration for accuracy. Unlike simple Euler integration, RK4 better preserves energy in oscillatory systems.
Chaos
The double pendulum exhibits deterministic chaos: the motion is fully determined by initial conditions, yet impossible to predict long-term due to extreme sensitivity. Two pendulums starting with nearly identical positions will diverge exponentially over time.