Double pendulum · glm-4.7-flash

Output generated Cost / run $0.0116 Runtime 189.3s

Prompt for this run

Build a planar double-pendulum simulation on a canvas with a colorful fading trail at the lower bob. Use HTML, CSS, and JavaScript as needed. Write it to `index.html` in the current working directory using your file tools. Inline your own CSS and JavaScript. Do not use external assets, CDNs, or network requests. No third-party libraries are provided. Required behavior: - Model two point masses connected by massless rigid rods under gravity, without damping or external driving. Use the coupled equations of motion rather than independent sine-wave animations. - Use equal masses and rod lengths, gravity of 9.81 in consistent simulation units, initial angles of 120 and -10 degrees measured from downward vertical, and zero initial angular velocities. - Draw a fixed pivot, both rods and bobs, and a bounded fading trail of the lower bob. Keep both rod lengths constant and the pendulum visible. - Provide Pause/Resume and Reset. Reset restores the stated initial conditions and clears the trail. - Use a numerically stable integration scheme and time step so ordinary playback does not explode or freeze. Verify: Sample the pendulum at several moments to inspect coupled motion, connected rods, and the lower-bob trail. Pause and resume to check the motion control, then reset and confirm that the starting pose and cleared trail return.

⛨ sandboxed Open in new tab ↗

TAKE IT TO CODE

A single self-contained HTML file — copy it and open in any browser, or reuse the prompt with your own pi-agent run.

RUN

modelglm-4.7-flash

vendorZ.ai · GLM

harnesspi 0.87.1 · browser iteration

generated2026-10-09

METRICS

runtime189.3s

tokens in230k

tokens out9.7k

cached122k

cost / run$0.0116

Transcript · 45 messages

system

user

assistant

thinking

Let me write a double pendulum simulation with RK4 integration, canvas rendering, trail, pause/resume, reset buttons.

Physics: θ1, θ2 from downward vertical, equal masses m, lengths l1=l2=l=1, g=9.81.

Standard equations:
θ1'' = (-g(2m1+m2)sinθ1 - m2 g sin(θ1-2θ2) - 2 sin(θ1-θ2) m2 (θ2'^2 l2 + θ1'^2 l1 cos(θ1-θ2))) / (l1 (2m1+m2 - m2 cos(2θ1-2θ2)))

With m1=m2=m, l1=l2=l: divide numerator/denominator by m:
θ1'' = (-3g sinθ1 - g sin(θ1-2θ2) - 2 sin(θ1-θ2)(θ2'^2 + θ1'^2 cos(θ1-θ2))) / (l(3 - cos(2θ1-2θ2)))... let me verify: denominator l1(2m1+m2 - m2 cos(2θ1-2θ2)) = l*m(3 - cos(2Δ)). Numerator: -g(3m) sinθ1 - m g sin(θ1-2θ2) - 2 sin(Δ) m (θ2'^2 l + θ1'^2 l cosΔ). Factor m: [-3g sinθ1 - g sin(θ1-2θ2) - 2 sinΔ (l θ2'^2 + l θ1'^2 cosΔ)] / [l(3-cos2Δ)].

θ2'' = (2 sinΔ (θ1'^2 l1 (m1+m2) + g(m1+m2)cosθ1 + θ2'^2 l2 m2 cosΔ)) / (l2 (2m1+m2 - m2 cos2Δ))
With equal: = 2 sinΔ (2 θ1'^2 l + 2g cosθ1 + θ2'^2 l cosΔ) / (l (3 - cos2Δ)).

RK4 with dt = 1/240 s substeps per frame, using real elapsed time clamped.

Trail: store positions of lower bob, cap ~500 points, draw with fading alpha and hue rotation for color. Use globalCompositeOperation maybe; simpler: draw trail segments with alpha proportional to index and hue.

Controls: Pause/Resume toggle button, Reset button. Also maybe speed but not required.

Write index.html then run in browser to verify.

→ write index.html (3272 bytes)

toolResult

Successfully wrote to index.html

assistant

thinking

I accidentally wrote a messy draft with placeholder code. Let me rewrite the whole file cleanly with the full correct implementation.

I made a draft with placeholder code by mistake — let me rewrite the whole file cleanly.

→ write index.html (8611 bytes)

toolResult

Successfully wrote to index.html

assistant

thinking

Wait — gravity sign check. Angles measured from downward vertical. With y downward positive (canvas), bob position y = L*cos(θ). Downward is +y, gravity potential U = -m g y? Actually with y downward, potential energy U = -m g y (higher y = lower). Let me check the equations of motion convention.

Standard derivation: θ from downward vertical, position x = L sinθ, y = -L cosθ (y upward positive). Then equations I used are the standard ones. In canvas, y_screen = PIVOT_Y + L cosθ, which means screen-down = physics-down. So physics y_up = -cosθ. My energy readout: y1s = -L cosθ1 — correct for upward-positive. Good.

The EOM: standard double pendulum (from downward vertical):
θ1'' = [−g(2m1+m2) sinθ1 − m2 g sin(θ1−2θ2) − 2 sin(θ1−θ2) m2 (θ2'² L2 + θ1'² L1 cos(θ1−θ2))] / [L1(2m1+m2 − m2 cos(2θ1−2θ2))]

With m1=m2=1: denominator = L1(3 − cos2Δ). Numerator = −3g sinθ1 − g sin(θ1−2θ2) − 2 sinΔ (θ2'² L2 + θ1'² L1 cosΔ). ✓ matches a1.

θ2'' = [2 sin(θ1−θ2) (θ1'² L1 (m1+m2) + g(m1+m2) cosθ1 + θ2'² L2 m2 cos(θ1−θ2))] / [L2(2m1+m2 − m2 cos2Δ)]

With equal masses: numerator = 2 sinΔ (2θ1'²L1 + 2g cosθ1 + θ2'² L2 cosΔ). ✓ matches a2.

Energy readout KE: lower bob velocity squared = (L1 ω1)² + (L2 ω2)² + 2 L1 L2 ω1 ω2 cos(θ1−θ2). ✓

Sanity check energy at start: θ1=120°, θ2=−10°, ω=0. KE=0. y1s = −cos120° = 0.5. y2s = 0.5 − cos(−10°) = 0.5 − 0.9848 = −0.4848. PE = 1*9.81*0.5 + 1*9.81*(−0.4848) = 4.905 − 4.756 = 0.149. Total E ≈ 0.149 J (per unit). It will be displayed and should stay constant — good verification.

One issue: the advance() function — the accumulator logic is awkward. dtReal is typically 1/60 = 0.01667, steps = round(0.01667*480)=8. Fine. If steps >= 1 we don't use the accumulator; fractional drift is dropped but clamping handles big frames. That's acceptable but slight timing irregularity. Actually with round, average keeps up. But small frames (high refresh 144Hz): dt=0.00694, steps = round(3.33)=3 → slightly fast. Better to always use accumulator. Let me simplify: always accumulate.

Also energy readout should include check. Let me fix advance():

function advance(dtReal) {
  dtReal = Math.min(dtReal, 0.05);
  advance.acc = (advance.acc || 0) + dtReal;
  let steps = Math.floor(advance.acc / DT);
  if (steps > 2400) { steps = 2400; advance.acc = 0; }
  else advance.acc -= steps * DT;
  ...
}

Also trail sampling every 4 substeps: with dt clamp, fine.

Also there's a bug: energy drift check — RK4 at dt=1/480 is very accurate; fine.

One more consideration: "bounded fading trail" — I cap at 600 points ✓, fading alpha ✓, colorful ✓.

Let me apply the advance fix and then test in browser.

Now let me simplify the time-step accumulator logic to be more robust:

→ edit

{"path":"index.html","edits":[{"oldText":"function advance(dtReal) {\n  dtReal = Math.min(dtReal, 0.05);        // clamp long frames / tab s

toolResult

Successfully replaced 1 block(s) in index.html.

assistant

Now let me verify the physics numerically before loading it in the browser:

→ bash

{"command":"node -e \"\nconst G=9.81, L1=1, L2=1, DT=1/480;\nfunction deriv(s){const d=s.t1-s.t2, sd=Math.sin(d), cd=Math.cos(d);\nconst den

toolResult

t=0 E= 0.149036 theta1= 120.0 theta2= -10.0 t= 1s E= 0.149036 E-E0= -3.85e-8 theta1= -87.0 theta2= 14.4 |o1|= 1.98 |o2|= 4.09 t= 5s E= 0.149036 E-E0= -7.41e-8 theta1= -44.2 theta2= 150.4 |o1|= 1.45 |o2|= 1.62 t=10s E= 0.149036 E-E0= -2.27e-7 theta1= 13.7 theta2= -183.5 |o1|= 0.44 |o2|= 3.89 t=30s E= 0.149035 E-E0= -1.09e-6 theta1= -74.5 theta2= -1002.1 |o1|= 2.16 |o2|= 1.13 t=60s E= 0.149034 E-E0= -2.37e-6 theta1= -37.0 theta2= -2483.1 |o1|= 4.92 |o2|= 0.47 finite after 60s: true

assistant

thinking

Physics is stable: energy conserved to ~1e-6 over 60s, chaotic motion with full swings. Rod constraints are inherent to the angle formulation. Now run in the browser and check visuals.

Physics verified: energy conserved to ~1e-6 over 60 s of chaotic motion, angles evolve in a genuinely coupled way (θ2 winds up past −1000°), values stay finite. Now let me check the rendering in the browser:

→ browser {"action":"reload"}

toolResult

{"console":["warning: An iframe which has both allow-scripts and allow-same-origin for its sandbox attribute can escape its sandboxing."],"dom":"Planar Double Pendulum (RK4)\nPause\nReset\nEqual masses & rods, g = 9.81, no damping — chaotic coupled motion","images":"model does not support images"}

assistant

→ browser {"action":"screenshot"}

toolResult

assistant

→ browser {"action":"inspect"}

toolResult

assistant

thinking

The browser tool doesn't give me screenshots I can see. Let me inspect the DOM more deeply — maybe it returns canvas pixel data? Let me try evaluating JS state via a script injected... The browser tool only supports listed actions. I can't run arbitrary JS. But I can check behavior indirectly:

1. Initial state check — I can't read canvas pixels directly. But I could add a debug hook? Better approach: I can verify code correctness by review, and use motion/screenshot actions which unfortunately return "model does not support images."

Let me at least check that the canvas element exists and buttons work via inspect after clicking. Click Pause button and see DOM change (button text "Resume"). The DOM output shows text content, so clicking pause should show "Resume".

The browser tool can't show me images, but I can verify interactive behavior through the DOM. Let me test Pause → Resume → Reset:

→ browser {"action":"click","x":394,"y":694}

toolResult

assistant

→ browser {"action":"inspect"}

toolResult

assistant

thinking

The DOM text is static (buttons' live textContent not reflected, or the click missed the button). The DOM snapshot only shows static HTML, not runtime state. I can't observe runtime changes through this tool's text output. To make verification possible, I could expose a debug API (window.__sim) and add an on-page status readout that includes paused state and trail length — those appear in DOM? No — DOM snapshot appears to be static HTML at load, not live text.

Alternative: use the console channel — the tool reports console messages. I could have the sim log state changes to console. Let me add minimal console logging on pause/reset/periodic state, reload, click, and check the console output in the browser tool response.

The DOM snapshot appears static, so I'll expose runtime state via console logging to verify behavior. Let me add a small debug hook:

→ edit

{"edits":[{"newText":"  const dbg = window.__sim = window.__sim || {};\n  dbg.paused = paused; dbg.trailLen = trail.length;\n  dbg.state = {

toolResult

Successfully replaced 4 block(s) in index.html.

assistant

thinking

Wait — in the draw() function, the debug assignment references `ke + pe` which are defined before. Let me check the placement — the oldText was after drawBob? No, in draw() after energy computation. The `dbg.info` line is fine. Also dbg.state uses `ke + pe` which is in scope. Good.

Also note: the E readout line ordering — the debug block replaced the fillText; ke and pe are computed above it. Good.

Now reload, wait, and check console.

→ browser {"action":"reload"}

toolResult

{"console":["warning: An iframe which has both allow-scripts and allow-same-origin for its sandbox attribute can escape its sandboxing.","log: [sim] paused=false trailLen=0 E=0.14904 θ1=120° θ2=-10°"],"dom":"Planar Double Pendulum (RK4)\nPause\nReset\nEqual masses & rods, g = 9.81, no damping — chaotic coupled motion","images":"model does not support images"}

assistant

thinking

Initial state confirmed: E=0.14904 (exactly the analytic value 0.149036), θ1=120°, θ2=-10°, running (paused=false). Trail length 0 at t=0 — correct since trail starts empty and builds up.

Wait for a few seconds and check motion + trail growth.

Initial state confirmed: θ1=120°, θ2=−10°, E=0.14904 (matches the analytic value exactly), running. Let me wait a bit and check the coupled motion and trail:

→ bash {"command":"sleep 4"}

toolResult

(no output)

assistant

→ browser {"action":"inspect"}

toolResult

assistant

thinking

Excellent:
- Motion is happening: θ1, θ2 evolve in coupled fashion, different from each other.
- Trail grows to the cap of 600 and stays bounded. ✓
- Energy constant at 0.14904 throughout. ✓

Now test Pause: click the Pause button. Button location: canvas is 640 high + header ~ padding. Layout: wrap padding 14, h1 (~24px), gap 10, canvas 640, gap 10, buttons at y ≈ 14+24+10+640+10+half button (~38/2=19) ≈ 717. Earlier I clicked at (394,694) — may have missed. Buttons: canvas centered horizontally, width 900 → x from 150 to 1050... wait viewport is 1200 wide. Pause button around x=520? "Pause" button centered? The controls div is flex centered; two buttons with gap: Pause ~90px wide, Reset ~80px. Total ~180px, centered at 600. Pause spans ~510-600, Reset ~610-690. y ≈ 14+24+10+640+10+19 = 717.

Let me click Pause at (550, 717).

Motion verified: coupled angles diverge (θ1=66°/θ2=−22° → θ1=104°/θ2=15°), trail grows to the 600-point cap and stays bounded, energy constant at 0.14904. Now testing Pause:

→ browser {"action":"click","x":550,"y":717}

toolResult

assistant

thinking

Pause worked: DOM now shows "Resume" (button text updated) and console logs "[sim] PAUSED". Let me confirm the state freezes (wait 3s, check log shows paused=true and same angles), then resume, then reset.

Pause works — button now reads "Resume", sim logs PAUSED. Let me confirm the state is frozen:

→ bash {"command":"sleep 3"}

toolResult

(no output)

assistant

→ browser {"action":"inspect"}

toolResult

assistant

thinking

State frozen: θ1=-111°, θ2=-17° identical across 4 log entries while paused. Now click Resume (same button position), wait, confirm motion resumes, then click Reset and verify initial conditions + cleared trail.

State frozen (identical θ1=−111°, θ2=−17° across four samples). Now Resume, then Reset:

→ browser {"action":"click","x":550,"y":717}

toolResult

assistant

→ bash {"command":"sleep 3"}

toolResult

(no output)

assistant

→ browser {"action":"click","x":655,"y":717}

toolResult

{"console":["warning: An iframe which has both allow-scripts and allow-same-origin for its sandbox attribute can escape its sandboxing.","log: [sim] paused=false trailLen=0 E=0.14904 θ1=120° θ2=-10°","log: [sim] paused=false trailLen=273 E=0.14904 θ1=66° θ2=-22°","log: [sim] paused=false trailLen=522 E=0.14904 θ1=104° θ2=15°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-40° θ2=146°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-16° θ2=31°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-54° θ2=-151°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=45° θ2=142°","log: [sim] PAUSED trailLen=600","log: [sim] paused=true trailLen=600 E=0.14904 θ1=-111° θ2=-17°","log: [sim] paused=true trailLen=600 E=0.14904 θ1=-111° θ2=-17°","log: [sim] paused=true trailLen=600 E=0.14904 θ1=-111° θ2=-17°","log: [sim] paused=true trailLen=600 E=0.14904 θ1=-111° θ2=-17°","log: [sim] paused=true trailLen=600 E=0.14904 θ1=-111° θ2=-17°","log: [sim] paused=true trailLen=600 E=0.14904 θ1=-111° θ2=-17°","log: [sim] RESUMED trailLen=600","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-46° θ2=-22°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-110° θ2=-341°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-60° θ2=-45°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-84° θ2=-355°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-83° θ2=-307°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-69° θ2=-269°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-90° θ2=-293°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=5° θ2=-75°","log: [sim] RESET trailLen=0 state={\"t1\":2.0943951023931953,\"t2\":-0.17453292519943295,\"o1\":0,\"o2\":0,\"energy\":0.1490359429502357}"],"dom":"Planar Double Pendulum (RK4)\nPause\nReset\nEqual masses & rods, g = 9.81, no damping — chaotic coupled motion","images":"model does not support images"}

assistant

thinking

All verified:
- Resume works: motion continues after pause (angles change again).
- Reset works: state restored exactly (t1=2.0944 rad = 120°, t2=−0.17453 rad = −10°, ω=0,0, energy 0.149036) and trail cleared (trailLen=0).

Let me confirm the sim is running again after reset with the initial pose, and check one more log entry showing it re-evolving.

Everything verified. One final check that the sim re-evolves from the reset pose:

→ bash {"command":"sleep 3"}

toolResult

(no output)

assistant

→ browser {"action":"inspect"}

toolResult

{"console":["warning: An iframe which has both allow-scripts and allow-same-origin for its sandbox attribute can escape its sandboxing.","log: [sim] paused=false trailLen=0 E=0.14904 θ1=120° θ2=-10°","log: [sim] paused=false trailLen=273 E=0.14904 θ1=66° θ2=-22°","log: [sim] paused=false trailLen=522 E=0.14904 θ1=104° θ2=15°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-40° θ2=146°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-16° θ2=31°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-54° θ2=-151°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=45° θ2=142°","log: [sim] PAUSED trailLen=600","log: [sim] paused=true trailLen=600 E=0.14904 θ1=-111° θ2=-17°","log: [sim] paused=true trailLen=600 E=0.14904 θ1=-111° θ2=-17°","log: [sim] paused=true trailLen=600 E=0.14904 θ1=-111° θ2=-17°","log: [sim] paused=true trailLen=600 E=0.14904 θ1=-111° θ2=-17°","log: [sim] paused=true trailLen=600 E=0.14904 θ1=-111° θ2=-17°","log: [sim] paused=true trailLen=600 E=0.14904 θ1=-111° θ2=-17°","log: [sim] RESUMED trailLen=600","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-46° θ2=-22°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-110° θ2=-341°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-60° θ2=-45°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-84° θ2=-355°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-83° θ2=-307°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-69° θ2=-269°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-90° θ2=-293°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=5° θ2=-75°","log: [sim] RESET trailLen=0 state={\"t1\":2.0943951023931953,\"t2\":-0.17453292519943295,\"o1\":0,\"o2\":0,\"energy\":0.1490359429502357}","log: [sim] paused=false trailLen=197 E=0.14904 θ1=-29° θ2=-114°","log: [sim] paused=false trailLen=440 E=0.14904 θ1=-32° θ2=43°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-38° θ2=221°","log: [sim] paused=false trailLen=600 E=0.14904 θ1=-81° θ2=-54°"],"dom":"Planar Double Pendulum (RK4)\nPause\nReset\nEqual masses & rods, g = 9.81, no damping — chaotic coupled motion","images":"model does not support images"}

assistant

thinking

All checks pass. The debug logging is benign (console only, every 2s) — I could remove it, but it's harmless and useful. Actually, for a clean artifact, maybe reduce it? It's fine to leave — it doesn't affect the app. I'll leave it; it's a small dev aid. Actually the task says "Fix clear problems" — logging isn't a problem. I'll leave it in but it's a minor cosmetic choice. Let me keep it.

Final report.

All checks pass. The simulation is complete and verified in `index.html`. ## What was built A single self-contained `index.html` with: **Physics (real coupled dynamics, not sine-wave animation)** - Full coupled equations of motion for equal masses/rods (angles measured from downward vertical), integrated with **classic RK4** at a fixed **dt = 1/480 s** sub-step, with a frame-time accumulator and clamping (max 0.05 s/frame, max 2400 sub-steps) so long frames or background tabs can't explode or freeze it. - g = 9.81, m₁ = m₂ = 1, L₁ = L₂ = 1, θ₁(0) = 120°, θ₂(0) = −10°, ω₁ = ω₂ = 0. - No damping or driving — energy is conserved. **Rendering** - Fixed pivot, both rigid rods drawn as connected segments pivot→bob1→bob2 (rod lengths constant by construction of the angle formulation), gradient-shaded bobs. - Lower-bob trail: sampled every 4 sub-steps, capped at 600 points, drawn with alpha fading toward the tail and a hue sweep (blue→purple→pink) for color. **Controls**: Pause/Resume toggle and Reset (restores exact initial conditions and clears trail). ## What I verified (via console/state instrumentation + DOM) - **Numerical stability**: offline RK4 test — energy constant at 0.149036 with drift ~2×10⁻⁶ after 60 s of chaotic motion; θ₂ wound past −2400° while remaining finite. - **Coupled motion**: live samples showed θ₁/θ₂ evolving independently of each other (e.g. 66°/−22° → 104°/15° → −54°/−151°) with constant E = 0.14904. - **Trail**: grows from 0 → 600 and stays bounded at the cap. - **Pause/Resume**: paused state froze angles at −111°/−17° across 4+ samples; resume continued from exactly that state; button label toggles correctly. - **Reset**: state returned to exactly t₁=2.0944 rad (120°), t₂=−0.17453 rad (−10°), ω=0, trail cleared to 0, and motion re-evolved from the initial pose afterward. **Unverified visually**: the browser tool couldn't return screenshots, so the exact pixel appearance (trail color gradient, bob shading, grid glow) is unconfirmed by eye — though the draw code paths all executed without console errors.