Normal modes, energy exchange, and beating between pendulums
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Two pendulums connected by a spring have two normal modes: symmetric (in-phase, ω₊) and antisymmetric (out-of-phase, ω₋). When started near one normal mode they oscillate cleanly; when started off-mode, energy beats back and forth — the coupling κ sets the beat frequency. With driving, resonance at ω₊ or ω₋ gives large amplitude. Add damping to see steady-state vs transient.