Double pendulum

📋 Teacher notes

• •The double pendulum is one of the simplest systems that exhibits deterministic chaos — the motion is governed by exact equations, yet long-term behaviour is effectively unpredictable.
• •Chaos arises from the nonlinearity of the equations, NOT from randomness or noise. Given the same initial conditions, the trajectory is perfectly repeatable.
• •The system is extremely sensitive to initial conditions: a change of 0.01° in starting angle produces a completely different trajectory after just a few seconds — the "butterfly effect".
• •Energy is conserved in the ideal (frictionless) case. The red/blue energy bar shows kinetic and potential energy trading back and forth. With damping, total energy slowly decreases.
• •For small angles (θ < 15°), the motion is approximately regular and periodic — the linearised equations decouple into two normal modes.
• •The trail of the second bob (orange) is the most vivid visual signature of chaos: ordered motion produces smooth, repeating curves; chaotic motion fills an irregular, tangled region of space.
• •In real experiments, friction and air resistance eventually damp the motion. The "Damped motion" preset shows how chaos gives way to regular decay as energy dissipates.
• •Q-factor and bandwidth are extracted live from the energy envelope. Q = ω/γ where γ is the exponential decay constant of total energy.