Double pendulum · mimo-v2.6-pro

Let me build the double pendulum simulation. Physics: standard double pendulum equations with m1=m2, l1=l2, g=9.81.

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))]
θ2'' = [2 sin(θ1-θ2)(θ1'^2 l1 (m1+m2) + g(m1+m2)cosθ1 + θ2'^2 l2 m2 cos(θ1-θ2))] / [l2(2m1+m2-m2cos(2θ1-2θ2))]

Use RK4 with dt small (e.g., 1/600 or so) with substeps per frame.

Trail: bounded array of points, fade alpha.

Let me write the file.

Initial angles: 120° and -10° from downward vertical. So θ measured from downward vertical; standard equations with θ measured from downward vertical work.

Set l1 = l2 = 1 (or scale in pixels). g = 9.81. Use units: lengths in meters, m=1.

Canvas ~ 800x600, scale px/m maybe 130, pivot at center-top-ish.

Implement:
- state [θ1, ω1, θ2, ω2]
- derivative function
- RK4 with fixed h = 1/480 s maybe; use accumulator with clamped frame delta.

Trail: array of {x,y}, max length e.g., 600 points; draw with hue cycling and alpha decreasing with age.

Pause/Resume button toggles. Reset restores angles and clears trail.

Also maybe draw trail as segments with lineWidth and color hsl. Bounded = fixed max length.

Let me write it, then run in browser.

Time check: do it efficiently. Write file now.

Design considerations: pendulum visible — canvas 1200x800 viewport but page may be smaller; use fixed canvas size e.g. 900x640 scaled with CSS max-width. Use devicePixelRatio maybe—keep simple but crisp: handle DPR.

Let me write.

deriv(t, y):
const [t1, w1, t2, w2] = y; m1=m2=1, l1=l2=1, g=9.81
delta = t1 - t2
den = 2*m1 + m2 - m2*Math.cos(2*delta)  => 3 - cos(2delta)
a1 = (-g*(2*m1+m2)*Math.sin(t1) - m2*g*Math.sin(t1-2*t2) - 2*Math.sin(delta)*m2*(w2*w2*l2 + w1*w1*l1*Math.cos(delta))) / (l1*den)
a2 = (2*Math.sin(delta)*(w1*w1*l1*(m1+m2) + g*(m1+m2)*Math.cos(t1) + w2*w2*l2*m2*Math.cos(delta))) / (l2*den)

Energy sanity: fine.

Substep: dt fixed 0.002 s (0.5ms? no 2ms) — with RK4 that's fine and stable. Per frame at 60fps, 16.7ms => ~8 substeps. Use accumulator with max frame time 0.05s.

Actually to keep physics smooth, run fixed h = 1/500 = 0.002.

Trail points appended once per frame (or per substep? per frame is fine, but faster motion gives gaps; append per substep at coarser interval e.g. every 2 substeps). Let's append each frame using current bob position — simple and bounded at 900 points. Actually trail smoother if appended more often: append every substep (500/sec) => 900 points = 1.8 s of trail. Maybe max 1200 points = 2.4s. Fine. Let's append every other substep => 250/sec, 1000 points = 4s. Good.

Rendering trail: for i from 0..n-1, draw line segments from point i to i+1 with alpha = (i/n)*maxAlpha and hue = hueBase + i*0.15... Colorful: hue based on index/time.

Let me use ctx with 'lighter' maybe not; just standard alpha.

Also draw pivot: circle at pivot, maybe a mount bar.

Controls: buttons Pause/Resume (single toggle labeled), Reset. Maybe also a status display with time. Keep minimal.

Now write the code.

For DPR: canvas width/height attributes set to CSS size * dpr, ctx.setTransform(dpr,0,0,dpr,0,0).

Simpler: fixed logical 960x680, scale to fit container via CSS width:100%; height:auto; set canvas attributes 960*dpr x 680*dpr.

Let me write.

Trail hue: base hue that advances with time so colorful fading.

I'll write the file now.

Now run in browser.

Looks good: rods connected, pivot fixed, bobs drawn. But wait — at t=0.35s θ1 = 86.5° and θ2 = -17.1°. That seems plausible. But the upper bob appears at upper right while lower bob is below... Actually the drawing shows the upper bob (yellow) at right of the pivot, at slightly above the mount bar level? The rod from pivot to yellow bob goes up-right. θ1=86.5° from downward vertical means nearly horizontal — sin(86.5)=0.998 so x1 = pivot + 174.6, cos(86.5)=0.06 so y1 = pivot + 10.6. Yes, so yellow bob is right and slightly below pivot. In the image the yellow bob is slightly above the pivot line... y≈302 vs pivot 355. Hmm, that's above. That would mean cos θ1 negative. But readout says 86.5°.

Hmm wait — the readout at the moment of DOM snapshot may differ from the screenshot? The screenshot is taken at some time after. Actually the DOM text was captured with the frame at 0.35s but the screenshot may be later. Also, the trail — the trail should be visible. I see only a tiny orange/red trail segment near the lower bob. That's suspicious: trail had 88 points but at 0.35 s the pendulum had only just started moving... Actually trail shows only a tiny streak. Hmm — trail points at 250/sec for 0.35 s = 88 points, and the bob only moved a little at first... no, at 0.35s θ2=-17.1° and bob2 has moved. The bob position: x1 = 174.6+480 = 654.6... wait pivot x = 480 in logical (960/2). Screenshot is 1200 wide scaled: canvas displayed at width ~968 px for 960 logical → roughly 1:1.008. So pivot at ~603 in image (matches). Yellow bob at x≈773 image → logical ≈ (773-112)/1.008 ≈ 656. Yes matches x1=654.6. y: image 302 → logical (302-155)/1.008 ≈ 146? Hmm the canvas top is at image y≈155. Logical y1 = PIVOT_Y + 175*cos(86.5°) = 198 + 10.6 = 208. Image y = 155 + 208*1.008 ≈ 364. But yellow bob is at 302. Discrepancy — so the screenshot is at a different time than the DOM readout. Fine. That's likely: screenshot taken slightly later when θ1 is above horizontal.

Actually the screenshot may have been captured at a moment where the pendulum has swung up-left/up. Anyway motion looks correct.

The trail seems short though. At the moment of the screenshot, trail should have many points behind the pink bob. The pink bob's trail appears as a tiny orange streak. Hmm. Wait — the trail might be drawn correctly but only recent points are near the bob, and older points would be spread along the path. The path in 0.35s is short... but at time of screenshot, maybe only ~0.5s elapsed. The lower bob moved maybe 50px. Actually the streak looks ~15px. Hmm.

Wait, look more carefully: the pink bob is at image (723, 472) and there's a tiny orange streak just above-left of it around (718-730, 435-450). That's the trail. The trail length covers only a short distance because in 0.5 s the lower bob doesn't travel far... Actually it starts at 120° and swings; velocity grows. In 0.5s bob2 travels maybe 100+ px. Hmm.

Actually, one concern: the trail hue is based on index, and older points are dark/transparent (alpha = age²·0.85). Older points nearly invisible. That's the "fading" behavior — fine.

Let me let it run a few seconds and take a screenshot to see the trail flourish. Let me use motion or just wait. I can use browser "motion" to see frames? Let me do a screenshot after some delay — I can run a click or just take another screenshot later (each tool call takes time anyway). Let me take a couple screenshots.

Also I want to verify pause/resume and reset. Plan:
1. Take screenshot after ~3-5 s to see trail.
2. Click Pause button, verify status "Paused" and time stops (readout twice).
3. Click Resume, verify motion continues.
4. Click Reset, verify time ~0, angles 120/-10, trail cleared (trail count 1).

Buttons are in DOM; browser click at coordinates. Toggle button is at around x=148,y=463 in the image (button "Pause"). Let me inspect the DOM to get positions... The screenshot shows the controls are below the fold (viewport 1200x800; the page shows controls at bottom partially? Actually the screenshot shows only the stage, controls not visible). The DOM has them. I can scroll down first.

Let me click via coordinates: better to scroll then screenshot.

Alternatively I could run checks via console eval — but the browser tool doesn't expose eval. Only inspect which dumps DOM/console. Hmm, "inspect" returns DOM text and console. That gives readouts (time, angles, trail count) which is exactly what I need to verify pause/reset numerically.

Sequence:
- screenshot now-ish to see trail growth.
- inspect → check time increasing.
- scroll down, screenshot to see buttons.
- click Pause at its coordinates; inspect → time frozen (two inspects).
- click Resume; inspect → time increases.
- click Reset; inspect → time 0.00, angles 120/-10, trail 1; screenshot to confirm cleared trail.

Let me do it.

62 s of sim time has elapsed (many tool calls between). The trail is colorful and fading, bounded at 1400 points. The pendulum looks good: rods connected, pivot fixed, trail attached to lower bob.

Physics sanity: over 62 s of conservative double pendulum with 120° start, it stays bounded (rods remain length ~constant visually). Good — no explosion.

One check: rod lengths constant — visually yes.

Now test Pause/Resume and Reset. Scroll down to see buttons.

Now the lower bob is above (θ2 ≈ -176.9°), the trail is colorful and fading. Coupled motion evident.

Now scroll further to see the control buttons and click Pause.

Controls are visible: Pause button at (172, 742), Reset at (263, 742). Click Pause.

Paused: status "PAUSED", button "Resume". Now verify time is frozen — inspect twice and compare.

Time frozen at 119.86 s across two samples, angles identical, trail unchanged. Now resume.

Resumed — RUNNING, time advancing again (119.94 s and moving). Now Reset.

Reset cleared the trail (only 17 short points just after restart) and restarted near 120°/-10°. Good.

One more precise check: pause immediately after reset to confirm exact initial pose (θ1=120.0°, θ2=-10.0°, time small). Let me pause quickly and inspect. Click Pause at (172,742).

Actually the readout right after reset should be exact at t=0; but the render continued. To verify the exact initial conditions I can pause right away and check readout near 0. But even with pause, time has already advanced slightly. However reset() sets exact values and the readout showed 0.07 s withθ1 118.8 (which is consistent with physics starting at 120 with zero velocity — it decreases). That's strong evidence.

Let me do one final check: click Pause and read values, confirming near120/-10 with time frozen. That validates reset pose + pause interplay.

Actually, one thing I could improve: verify energy stability — already evident from 120 s run with bounded amplitude.

Also, small detail: the trail "bounded" requirement — 1400 max, confirmed. Trail cleared on reset — confirmed (trail17 after reset vs 1400).

I think we're done. Let me do a final quick pause-inspect to show the near-initial pose frozen, then finish. Actually I already verified pause freezes. This is optional; do it quickly since it costs little.

The click took time (17 s of sim time passed between tool calls — real wall-clock in the sandbox), so the pose is advanced again. But earlier evidence: right after reset, readout was time 0.07/0.18 s with θ1 118.8°/110.6°, θ2 -10.6°/-13.8° — consistent with starting at 120°/-10° with zero velocity. That's sufficient verification of the initial pose.

Everything checks out. Reset also worked while running. Final file exists. Done — report.