Double pendulum · step-5-preview

Let me build this. ILS: build a single HTML file with canvas-based double pendulum simulation.

Requirements:
- Planar double pendulum, coupled equations of motion
- Equal masses, equal lengths, g=9.81
- Initial angles: θ1=120°, θ2=-10°, zero velocities
- Fixed pivot, rods, bobs, fading trail of lower bob
- Pause/Resume, Reset
- Stable integration (RK4)
- Verify in browser.

Let me write the code.

Equations of motion for 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)))
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 cos(2θ1-2θ2)))

With m1=m2=m, L1=L2=L, denominators simplify.

Let me pick L = 1.0, m=1. Use dt = 0.001 with RK4 substeps per frame (accumulate time). Base length scale: pendulum total length 2L = 2m... Need visible in canvas. Canvas 800x600. Let's scale such that total length ~ 300 px, i.e. L=150px, and physical L=1.0 → scale = 150 px per unit.

Pivot at (width/2, 180) maybe. Positions:
x1 = x0 + L sinθ1, y1 = y0 + L cosθ1 (since measured from downward vertical).
x2 = x1 + L sinθ2, y2 = y1 + L cosθ2.

Trail: store last N points of lower bob (bounded e.g. 300 points), draw with varying alpha (older = more transparent).

Buttons: Pause/Resume, Reset. Keyboard: space = pause, R = reset.

Let me write the file.

Write index.html.

There's a typo in CSS: `border: 1px solid #2a3category;` — bad declaration, harmless but let's fix it.

Also, dynamics sanity check: verify initial acceleration values with a quick node script to confirm the sim is coupled (not simple). Let me also check total length 2*SCALE = 340 px, pivot at 140, so lowest reach y=480 within 560 — but with angles, max y could be at bottom = 140+340=480 < 560. Good. Radius 12 so 492. Good.

Fix the CSS typo and run a quick node check of the integrator.

Energy drifts minimally over 200 s — stable. Motion is coupled and chaotic as expected (large θ2 windings).

Now verify in browser: load, screenshot, check console, interact with pause/reset.

App is running. Now check the trail rendering after some seconds, then pause, then reset.

Let me wait a bit — use motion or scroll? There's no wait. I can take a screenshot after doing some interactions. Let's click Pause first and inspect (which pauses), then screenshot to see trail.

Actually order: let trail build up. The inspect just now showed start pose correct (θ1≈120, θ2≈-10). Let me click pause, then screenshot to inspect trail and confirm motion freezes. Then click Reset and check θ1=120.0, θ2=-10.0, trail cleared.

Hmm — the click at (71,659) apparently didn't register a click (state still "running"). Probably coordinates are in viewport space but canvas scaled? The controls row is below the canvas at y≈659 in the 1200x800 viewport... The screenshot shows the page is scrolled? Actually the screenshot shows only the top portion — the buttons are at y≈659 which should be visible. But state didn't change. Maybe the click landed on the canvas region? No, canvas appears to be the big panel.

Wait — the DOM shows "Pause" button text at t=2.46 still running. The click may have missed. Let's inspect to get element positions? inspect gives DOM only. Let me try clicking with keyboard instead: press Space to pause, then screenshot.

Actually maybe the click did register but toggled twice? No.

Let's use key Space.

Paused works: state paused, t frozen at 5.37, trail visible with color gradient (fading). Note the trail shows a gap between the bob and recent trail? The trail is drawn behind the pendulum; the trail stops at the current bob position but the pendulum draws on top. There appears to be a short gap near the bob — probably because trail points are pushed once per frame and only from... hmm, the recent trail segment near the bob looks separated. Actually looking at the image: magenta bob at left, and red trail segment to below-right, with a gap of maybe a few pixels? Actually the trail ends at (x2,y2) each frame push, so the trail's last point should be at the bob. The trail stroke ends at bob position — the gap may be the line thickness/rounding. Actually looking at the first paused screenshot: the lower bob is the pink circle and the trail stops ~10px short. Hmm, could be because trail points are pushed after step but rendering uses state before? No — I push after stepping in the frame, then render. Order: step → push p → render. So last trail point = current bob. Unless... I push after the loop using bobPositions(state) computed after steps — yes correct.

Wait, but the visual gap: the trail's last point is at the bob center, and the bob is drawn on top of it with radius 12, so trail should be hidden under the bob. The screenshot shows the trail ending near the bob edge maybe slightly beyond. It's fine.

Hmm, actually there's a visible gap in the screenshot between the bob and trail start. Let me check: bob at x≈340,y≈305; trail red line goes from near there down-right. Actually the trail starts right at the bob's lower-left. Fine, it's probably a rendering artifact of the gradient alpha with f^2 → oldest fading.

Wait — actually a bug: I compute alpha as (0.06 + 0.75*f*f) where f = i/n, i from 1..n-1, so newest f≈1 alpha 0.81, oldest f≈0 alpha 0.06 — good fading.

Now resume and then reset to confirm. Let's press R to reset (from paused state) and check readout = 120.0/-10.0, t=0.00, trail cleared.

Reset works: starting pose restored and trail cleared. State remains paused (reset doesn't resume) — reasonable, but maybe better to keep pause state. It's fine; user can Resume.

Now resume via Space and let it run briefly to confirm motion continues.

Also verify the button clicks work with mouse (the earlier click at (71,659) didn't register — maybe because the button is located elsewhere). Let me check the layout: page is 1200x800 viewport, but the canvas is 860px wide inside a wrap max-width 900 → centered. The controls row would be around y≈? Canvas height 560 + header ~... the screenshot shows the whole page fits? The screenshot image includes buttons visible around 83% height of the image. The image is 1200x800. So the buttons are around y≈690-720 in image coords, but image scaling might not be 1:1. My click at y=659 hit the stage panel maybe. Let's click at approximately (71, 690)? Hmm, x=71 was based on the layout: wrap max-width 900 centered in 1200 → margins 150 each side; button at x≈160-200.

Better: use the browser tool to click on the Pause button by finding coordinates. I can estimate from the screenshot. The last screenshot shows the Pause button left edge around x≈180 in image pixels? The wrap is centered: (1200-900)/2 = 150, +16 padding = 166. Buttons at ~166..250, y? h1 top ~ margin 8 + 18 height ≈ 40; stage top ≈ 60; stage height = canvas height 560 scaled: canvas width 860 inside stage width 900-? wrap padding 16 → content 868 wide, canvas width:100% → 868; height auto = 868*(560/860)=565. So stage ≈ 60..625. controls at ≈ 639..675. So Pause button center ~ (207, 657). My earlier click at (71,659) was on the page background left of the button, not on it. That's why nothing happened.

Let me click at (207, 657) for Pause. But currently state is paused with Resume label. Let's do it.

Click on Pause button works (mouse), state running again.

One more check: verify the trail visibility at longer times and that rod lengths stay constant (they are computed analytically so yes). Also verify no console errors — console only shows the iframe sandbox warning (harness artifact, not our code).

Let me check the trail renders with a color fade over a longer run. Sample once more after a few seconds: pause and screenshot. Also confirm everything within canvas bounds (already saw). Let's do one final pause after ~10s of running. I can't wait explicitly; the click action advances frames. Let me press Space twice? Actions: each browser call yields a frame with time advanced. Let me do a scroll to force a frame / or just do Space pause now — it's only been ~0.05s.

Alternative: use "motion" action to advance frames? The motion action captures frames. Let me try key Space then screenshot — but time elapsed might be small.

Instead, do repeated screenshot actions to let time pass? Each call presumably advances time in the running page. Actually the page keeps running between tool calls regardless. The earlier sequence: t went 0.12 → 2.46 → 5.37 → 0.00(reset) → 0.05. So time passes between calls (each call maybe ~1-3 seconds of wall time). So now pausing would show a decent trail. Let's press Space.

Trail is visible with a nice gradient fade (cyan → magenta/yellow). Rods connected, bobs visible, trail bounded at 900 points.

Everything works. Final check: leave it running? It's fine. Also let's leave state running (press Space to resume) so the app is in a live state. Let me resume and confirm.

Also minor consideration: readout shows θ₂ ≈ 3521° mod → wrapped to -3521.93 mod 360... In JS, fmtDeg wraps into [-180,180], fine.

Let me resume via Space.