Draw Me Huygens' Pendulum Clock

Two forces, one rhythm

A pendulum is one of nature's simplest oscillators — two terms in an ODE, no exotic physics. Add a tiny damping term and you've got Huygens' clock; add a periodic kick at the right phase (an escapement) and the pendulum runs forever.

FORCE 1

gravity restores

−(g/L) · θ

The restoring torque is proportional to the displacement angle. For small angles it's linear; the period 2π·√(L/g) depends only on the rod length L and gravity g — not the bob's mass, not the amplitude.

FORCE 2

friction drains

−γ · ω

Air drag, pivot friction, escapement coupling. All combined into a single damping coefficient γ. A good clock keeps γ small — Huygens' first design lost less than a second per day, an order of magnitude better than the verge-and-foliot clocks before it.

THE FIX

escapement adds back

+ε per tick

The mainspring's energy reaches the pendulum through the escapement, which releases a small kick at each extreme. Net energy stays constant; the pendulum keeps a perfect rhythm; the gears advance a precise step each swing.

How Glyph drew it

Claude writes the JSON; Glyph integrates the 2D ODE with 4th-order Runge-Kutta over 1200 samples. Same spec → byte-identical SVG, every platform, every run.

The Glyph spec JSON

// pendulum-clock.json — damped harmonic oscillator
{
  "version": "glyph/0.1",
  "title": "Pendulum clock (Huygens 1656)",
  "data": {
    "trajectory": {
      "shape": "trajectory",
      // dθ/dt = ω (angular velocity)
      "dxdt": "y",
      // dω/dt = −(g/L)·θ − γ·ω
      // with g/L = 1, γ = 0.02
      "dydt": "-x - 0.02*y",
      "initial": { "x": 0.6,
                   "y": 0 },
      "time": { "min": 0,
                "max": 60,
                "samples": 1200 }
    }
  },
  "layers": [{
    "mark": "line",
    "encoding": {
      "x": { "field": "t" },
      "y": { "field": "x" }
    }
  }]
}

14 full swings over 60 seconds, with amplitude shrinking from 0.6 rad to ~0.45 rad — exactly the trajectory the ODE predicts. Try γ = 0 (perfect lossless pendulum), γ = 0.1 (audibly winding down), or γ = 0.5 (overdamped, no oscillation). View on GitHub.

Glyph compiler output SVG

Byte-stable across Ubuntu / macOS / Windows × Node 20 / 22. The exponential decay envelope is in the data — no separate "fit" curve drawn; the ODE produces it for free.

Your turn — prompts to try

Damped harmonic motion shows up everywhere — bridges, atoms, springs. Name your system and Claude can pick the ODE.

▸ Phase portrait

"Plot the same damped pendulum in phase space instead of time. x-axis = θ, y-axis = ω. The trajectory should spiral into the origin."

▸ Engineering

"Show me a Tacoma Narrows bridge oscillation — a forced damped harmonic oscillator with a periodic wind force. Find the resonance frequency where the amplitude grows unbounded."

▸ Chemistry

"Plot a Morse potential — the energy curve of a diatomic molecule like H₂. Show how a vibrating molecule is just a damped harmonic oscillator near the equilibrium bond length."

▸ Music

"Draw a plucked guitar string. Damped harmonic motion at the fundamental frequency, plus the first three overtones at integer ratios. Show the composite waveform."