Normal modes and chaotic trajectories — sensitive dependence on initial conditions
Physics: The double pendulum is a canonical chaotic system. Equations of motion (Lagrangian mechanics) couple θ₁ and θ₂ nonlinearly. For small angles, two normal modes exist: in-phase (both swing together) and out-of-phase (opposite). For large angles, the system is chaotic — nearby trajectories diverge exponentially (positive Lyapunov exponent). The phase portrait (θ₁ vs θ₂ here) shows the transition from regular KAM tori to chaotic regions. Energy is exactly conserved — the simulation uses symplectic RK4 integration.