Every pendulum has the same length L, so on a rigid support they would never influence each other. Here the support is a beam of finite mass M on a spring, free to slide horizontally — exactly the wall-mounted shelf Huygens used for his clocks. Each swinging bob pushes back on the beam (Newton's third law); the beam's resulting motion shifts the effective pivot point felt by every pendulum, coupling them all. This is derived from the real Lagrangian for N pendulums plus a translating support, not scripted:
θᵢ'' = -(g/L)sinθᵢ - (a/L)cosθᵢ - c·θᵢ'
a = [ Σᵢ m(g·sinθᵢcosθᵢ + L·c·θᵢ'cosθᵢ + L·θᵢ'²sinθᵢ) - k·x - cᵦ·ẋ ] / (M + Σᵢ m·sin²θᵢ)
a is the beam's shared horizontal acceleration, x its displacement, k its restoring spring, and c/cᵦ the pendulum and beam damping. Integrated with real RK4, this reproduces Huygens' 1665 observation: pendulums started at random phases exchange energy through the shared support and gradually settle into a common rhythm — in-phase or anti-phase depending on the coupling strength (lower support mass = stronger coupling = faster lock). Beam sway is drawn ×4 for visibility, exactly as strain is exaggerated in the gravitational-wave demo — the physics driving it is not.
Click any bob to kick it out of rhythm and watch the shared beam pull it back into sync.