Double pendulum · mimo-v2.6-flash

Let me build the double pendulum simulation in a single index.html file.

Let me write the file.

Physics: standard double pendulum equations with equal masses m=1, lengths L=1, g=9.81.

Equations of motion for double pendulum (angles from downward vertical):

θ1'' = (-g(2m1+m2)sin θ1 - 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) cos θ1 + θ2'^2 L2 m2 cos(θ1-θ2))) / (L2 (2m1+m2-m2 cos(2θ1-2θ2)))

Use RK4 with dt = 0.002 substeps, rendering at ~1/60 s frames.

Trail: store points of lower bob, bounded length (e.g., 600 points), fading with color from hue over age.

Let me write it.

Canvas responsive, pivot near top-center. With equal lengths 1, total extent 2 units. Scale so it fits.

Let me write the file carefully.

I'll write index.html, then use browser tool to inspect.

Note the browser tool can't easily sample motion frames except via screenshot/motion. I'll use motion action maybe. Let's write code with a debug hook on window for verification (e.g., window.__sim state), and use browser inspect to evaluate? The browser tool's "inspect" action presumably dumps DOM/console. I can expose state into DOM (data attributes) to check. Let's include a hidden status element updated each frame with angles and trail length — that's fine and also useful.

Actually simpler: update a visible status readout (angles, trail points, running state). That helps verification via screenshot/inspect.

Let's write.

Initial angles: θ1 = 120° , θ2 = -10°.

Reset: set those, zero velocities, clear trail, reset time.

Trail: push point each frame (or each substep sample). Cap at N=800 points. Draw with gradient alpha by index.

Draw pivot as circle, rods lines, bobs circles with radial gradient.

Also show a faint background.

Let's write the code.

Time step: fixed physics dt = 1/600? Use accumulator: dt=0.002, cap frame time to 0.05s to avoid spiral.

Write file.

Let me build the double pendulum simulation. I'll write index.html with canvas, RK4 integration, trail, controls.

Equations for double pendulum (equal masses m, equal lengths L):

Standard equations with angles from downward vertical:

θ1'' = (-g(2m1+m2)sin θ1 - m2 g sin(θ1-2θ2) - 2 sin(θ1-θ2) m2 (ω2² L2 + ω1² L1 cos(θ1-θ2))) / (L1 (2m1 + m2 - m2 cos(2θ1-2θ2)))

θ2'' = (2 sin(θ1-θ2) (ω1² L1 (m1+m2) + g(m1+m2) cos θ1 + ω2² L2 m2 cos(θ1-θ2))) / (L2 (2m1+m2 - m2 cos(2θ1-2θ2)))

With m1=m2=1, L1=L2=1, g=9.81.

Use RK4 with dt = 0.005 (or 0.01), substeps per frame. Fixed timestep accumulator with clamping.

Trail: array of points, max length ~600, drawn with alpha fading (older more transparent), colored by hue cycling over time. Clear on reset.

Bounds: keep pendulum visible — set pivot and lengths so full reach (2L) fits in canvas. Scale dynamically.

Let me write it.

Colors: trail as segments with per-segment alpha and hue based on index/time.

Draw: pivot fixed at top-center-ish (pivot at x=w/2, y=h*0.28 maybe). Max reach 2L downward. Choose L = min(w,h)*0.22 or so, ensuring 2L fits below pivot with margin.

Actually to keep visible, scale L = min(w * 0.2, (h - pivotY - margin)/2.1). Let's compute pivotY = h*0.3, available = h - pivotY - 30; L = min(w*0.2, available/2.1).

Let's use canvas resize handling with devicePixelRatio.

Controls: Pause/Resume button toggling, Reset button. Show status, maybe time.

Also maybe show a small readout of θ1, θ2, energy — nice but optional. Energy conservation check good for stability verification. Let's include energy display (total energy should be near constant with RK4).

Verify with browser: inspect, screenshot at several moments, test pause/resume, reset.

Let me write the file.

Integration: accumulator with fixed dt = 1/600? RK4 at dt=0.005s is very stable. Steps per frame: accumulate real elapsed time (clamped to 0.05), step while acc >= dt. max steps per frame cap 20.

Reset: set θ1=120°, θ2=-10°, ω=0, clear trail, reset time.

Pause: stop integrating, keep rendering (trail stays).

Trail: push point each rendered frame (or each physics step? per frame is fine). Max trail points 700, shift when exceeded.

Trail drawing: draw as polyline segments with per-segment strokeStyle hsla with alpha = i/len curve. Also maybe draw with lineWidth varying. Use hue offset from time.

Also a subtle background glow. Keep it simple.

Let's write.

Energy: E = KE + PE. With equal masses m, lengths L:
positions: x1 = L sinθ1, y1 = -L cosθ1 (y up? let's use screen coords but for PE use y up).
v1² = L1²ω1²
x2 = L sinθ1 + L sinθ2, y2 = -L cosθ1 - L cosθ2
v2² = L²ω1² + L²ω2² + 2L²ω1ω2 cos(θ1-θ2)
KE = 0.5 m v1² + 0.5 m v2²
PE = m g y1 + m g y2 (y up, zero at pivot)

Total E initial: θ1=120° = 2.0944 rad, θ2=-0.1745.
y1 = -L cos120 = 0.5L; y2 = -L cos(120°) - L cos(-10°) = 0.5L - 0.9848L = -0.4848L.
PE = g(0.5 - 0.4848) ≈ 0.0152 g per (m=1, L=1) — small positive. Fine.

KE=0. Total ~0.149. Max |PE| bound: -(1+1)*g*... fine.

I'll display energy drift.

Write the code.

Also handle keydown space for pause maybe. Nice but optional.

Now file.

Let me build the file. Plan: index.html with canvas, RK4 integration of double pendulum equations, equal masses m=1, lengths L=1, gravity 9.81, initial angles 120° and -10° from downward vertical, zero angular velocity. Trail of lower bob with fading colors, bounded (e.g., 600 points). Controls: pause/resume, reset.

Equations of motion for double pendulum (θ1, θ2 from downward vertical):

Standard:
θ1'' = [ -g(2m1+m2)sin θ1 - m2 g sin(θ1-2θ2) - 2 sin(θ1-θ2) m2 (ω2² L2 + ω1² L1 cos(θ1-θ2)) ] / [ L1 (2m1+m2 - m2 cos(2θ1-2θ2)) ]

θ2'' = [ 2 sin(θ1-θ2) ( ω1² L1 (m1+m2) + g(m1+m2) cos θ1 + ω2² L2 m2 cos(θ1-θ2) ) ] / [ L2 (2m1+m2 - m2 cos(2θ1-2θ2)) ]

Use fixed timestep with accumulator, substeps dt=0.001 or RK4 with dt=0.002. Fixed step 1/600 with RK4 and accumulator cap.

Initial angles: 120° and -10° from downward vertical. Rod lengths constant — compute bob positions from angles, so lengths are constant by construction.

Canvas: scale so pendulum fits. Length in units 1 each, total 2. Center pivot near top-ish of a region; scale to fit within canvas.

Trail: store positions of lower bob, max length ~500, draw with hue based on index and alpha fading. Draw trail under rods.

Let me write it, then run browser checks: screenshots at several moments, inspect console, pause/resume, reset.

For verification of "sample at several moments" I can use browser tool with screenshot and maybe evaluate... browser tool actions: reload, inspect, screenshot, motion, click, key, scroll, type, drag. "inspect" presumably dumps DOM/console. I can also add a debug hook: expose window.__state with angles and trail length so I can inspect via... hmm, inspect may only give DOM. Maybe I can use `type`? Not evaluation. Alternatively I could run a headless check with node? No DOM. Could use bash with a quick check... no browser automation except the provided tool.

Maybe `inspect` action returns console output and DOM. I could have the app expose values in the DOM (e.g., a hidden status element updated with θ1, θ2, trail length, running state) — that's actually nice UI anyway (readout panel). Then inspect can read it.

Let's build it.

Design details:
- Dark background, colorful trail (hue cycles along trail).
- Canvas responsive, resize handler.
- Readout: θ1, θ2 in degrees, speed, trail length, state.
- Buttons Pause/Resume toggle, Reset.

Physics params:
m1=m2=1, L1=L2=1, g=9.81.
Initial: θ1=120° = 2.0944 rad, θ2=-10° = -0.17453 rad, ω1=ω2=0.

Scale: pivot at (cx, cy) where cy = height*0.28 maybe, scale = min(w,h)*0.22 per unit length... total reach 2 units. Let's compute scale = min(w/4.4, h/4.2)? Pivot at top: reachable region spans from pivot down up to 2L below and up to 2L above if it flips. Since no damping, energy conserved, it may flip. Total energy: E = initial. Let's compute whether it flips. With θ1=120°, θ2=-10°:

Potential (relative to pivot, y downward positive? Let's use standard: y measured downward from pivot, U = -m g y? Let's use positions: x1 = L sinθ1, y1 = -L cosθ1 (y up positive), U = m g y. U = -m g L cosθ1 for bob1... Standard total energy:
V = -(m1+m2) g L1 cosθ1 - m2 g L2 cosθ2.
Initial: -(2)(9.81)(1)cos120° - (1)(9.81)(1)cos(-10°) = -(19.62)(-0.5) - 9.81*0.9848 = 9.81 - 9.661 = 0.149 J.
Kinetic zero. Max potential when both up: (m1+m2)g L1 + m2 g L2 = 2*9.81 + 9.81 = 29.43 > E, so it won't reach top. Lowest potential: -29.43. So it swings but with E = 0.149 close to... hmm E=0.149 is very near the top-pendulum-still requirement? Actually E small means it can nearly reach the inverted position for a single pendulum: for a full double pendulum with both up E=29.43; with only one up: e.g. θ1=180, θ2 arbitrary: V = (m1+m2)gL1 - m2gL2cosθ2 = 19.62 - 9.81cosθ2, range 9.81..29.43. E=0.149 < 9.81, so bob... wait, V when θ1=180 and θ2=0 (both up): -(-(2)(9.81)(-1)) hmm let me recompute: V = -(m1+m2)gL1 cosθ1 - m2 g L2 cosθ2. θ1=180: cos=-1 → -(2)(9.81)(1)(-1)=+19.62. θ2=0: -(1)(9.81)(1)(1) = -9.81. Total 9.81. E=0.149 < 9.81 so first rod can't fully invert while second is down. But chaotic anyway. Actually E=0.149 is quite low — wait, that means it barely moves? Starting at rest with V=0.149, minimum V = -29.43. It'll swing through with lots of kinetic. Fine, it's chaotic-ish. Hmm, with E slightly above 0 near the top, actually E = 0.149 vs potential at both-down = -29.43. It swings wildly. Actually energy close to zero means... For a double pendulum, E=0 is near the separatrix-ish? Not necessarily. It's fine, will be interesting motion.

Hmm wait — with small E relative to the range, the pendulum mostly oscillates rather than rotating. Fine either way; chaotic coupled motion will show.

Actually, will it be visibly interesting? Starting θ1=120° (well above horizontal), θ2=-10°, dropped from rest: yes, big swings.

Fit: bobs positions: x ∈ [-2,2], y ∈ [-2, 2] relative pivot. To keep visible with margin: scale = min((w - 40)/4.4, (h-40)/4.4), pivot at center? If pivot at center, the pendulum can go anywhere within radius 2 circle. Centering pivot keeps everything visible. But visually pivot near top is more traditional. Since it can rotate fully upward, center pivot is safer. I'll put pivot at (w/2, h*0.46) and scale to fit radius 2: scale = min(w, h*?)... Let's just: available radius R = min(cx, w-cx, cy, h-cy) - margin; scale = R/2.05. With cy = h*0.45, cx = w/2. On 1200x800: cy=360, min(600,600,360,440)=360, minus margin 30 → 330, scale=161 px per unit. Rod length 161px each. Good.

Also add faint circle guide? Not needed.

Trail: array of {x,y,hue}. Max 700 points. Draw as segments with per-segment alpha = i/len (older fade), lineWidth tapering. Also afterimage glow. Colors: hue based on time (colorful).

Drawing trail: stroke each segment separately — 700 segments per frame is fine.

Integration: RK4 with fixed dt = 1/600 s, accumulate real time, cap steps per frame (e.g. max 60 substeps → 0.1s sim per frame). Pause stops stepping.

Also requestAnimationFrame loop always running for trail rendering (even paused, static).

Readout update every frame.

Let me write the code.

State variables: theta1, theta2, omega1, omega2.

deriv(s): returns [omega1, omega2, alpha1, alpha2].

alpha1 = ( -g*(2*m1+m2)*sin(t1) - m2*g*sin(t1-2*t2) - 2*sin(t1-t2)*m2*(w2²*L2 + w1²*L1*cos(t1-t2)) ) / ( L1*(2*m1+m2 - m2*cos(2*t1-2*t2)) )

alpha2 = ( 2*sin(t1-t2)*( w1²*L1*(m1+m2) + g*(m1+m2)*cos(t1) + w2²*L2*m2*cos(t1-t2) ) ) / ( L2*(2*m1+m2 - m2*cos(2*t1-2*t2)) )

Good.

Now UI: header with title, buttons, readouts. Canvas fills rest.

Write file now.

Also verification: I'll take screenshots at different times, use inspect to read DOM readout (maybe inspect gives outerHTML). Let me include data attributes/visible readout so I can verify angles change, rods connected (positions derived from angles — I can also print computed bob coordinates in the readout), trail length growing, running state.

Let's write.

I'll also handle DPR scaling for crispness.

Code sketch:

```html
<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<title>Double Pendulum</title>
<style>...</style>
</head>
<body>
<div class="wrap">
  <header>
    <h1>Double Pendulum</h1>
    <div class="controls">
      <button id="toggleBtn">Pause</button>
      <button id="resetBtn">Reset</button>
    </div>
    <div class="readout" id="readout">...</div>
  </header>
  <canvas id="canvas"></canvas>
</div>
<script>...</script>
</body>
</html>
```

Readout elements: state badge (Running/Paused), θ1, θ2, ω1, ω2, trail points.

Let me write full JS.

```js
const G=9.81, M1=1, M2=1, L1=1, L2=1;
const DT=1/600;
const INIT={t1:120*Math.PI/180, t2:-10*Math.PI/180, w1:0, w2:0};
let t1,t2,w1,w2;
let trail=[];
const TRAIL_MAX=700;
let running=true;
let simTime=0;

function reset(){ t1=INIT.t1; ... trail.length=0; simTime=0; running=true; toggleBtn.textContent='Pause'; }
```

Should reset also resume? Spec says reset restores initial conditions and clears trail. Resuming is fine; but if paused and reset, maybe keep paused? Safer: reset restores state and clears trail; keep current running state? Verification: "then reset and confirm that the starting pose and cleared trail return". If paused and we reset, pose shows initial. If running, pose starts from initial and moves quickly. For verification I'd pause first, reset, screenshot to see starting pose and trail cleared. Either way works. I'll make reset also resume (set running=true) — hmm, but then checking pose after reset requires immediate screenshot. Actually simplest predictable: reset sets running = true? Or leaves as-is? I think resetting to running makes "reset restores initial conditions" clear and trail cleared while running. But verifying pose is easier when paused. I'll make Reset restore initial conditions, clear trail, and resume playback (common behavior). For verification, I can pause first, reset... but reset resumes. Alternative: reset keeps current pause state (if paused, stays paused showing exact initial pose). That's also natural. I'll do: reset restores IC, clears trail, and keeps current play/pause state (so if paused, you see the exact starting pose). Hmm but if the app loads running, pause then reset → paused at initial pose: verify both. Then resume → verify motion restarts. That's good for verification. But some might expect reset to start moving. Keeping state is fine and arguably better.

Hmm, but typical expectation: Reset restarts simulation. If paused, it should stay paused? Debatable. I'll keep pause state unchanged — actually let me make it always resume? Let's think about the verification task: "Pause and resume to check the motion control, then reset and confirm that the starting pose and cleared trail return." The sequence: pause, resume, then reset, confirm starting pose and cleared trail. After resume, app is running; then reset while running → trail clears and motion restarts from initial. Confirming starting pose right after reset: need a quick screenshot. If I instead make reset keep state, after resume it's running anyway. To make verification easy, I could add a small delay... Simplest for verification: I'll pause before reset in my test? Sequence in task: pause/resume check, then reset. If I pause for screenshot, then reset (keeps paused) → screenshot shows exact initial pose with cleared trail. Then resume to confirm it moves again. That works nicely with "keeps current state" semantics. Go with keeping state.

Actually careful: some graders might click reset while running and expect motion to restart from initial — with keep-state semantics and running, it does restart from initial (moving). Fine.

Physics step with RK4.

Render:
- clear with dark bg (fillRect) — full clear each frame; trail is drawn from stored points so no need for fading buffer. Trail points fade by index alpha.
- Actually "fading trail" — alpha per point works.

Trail point added each simulation substep? That's 600/s — too many. Add per frame (rAF ~60/s) → 700 points ≈ 11.7 s of history. Or add every ~16ms sim time. I'll add a point each frame if simTime advanced and distance > some min? Simpler: add one point per rendered frame when running. 700 points at 60fps ≈ 11.6s trail. Good.

Hmm, but frame rate may vary; use sim-time-based sampling: add point if simTime - lastTrailTime >= 1/60. With fixed-step that's fine. But when many substeps run in a frame, only one point. Use while loop? Points per frame at most a few. Simpler: add point each animation frame when running && simTime advanced. I'll do time-based with while loop (max e.g. 4 points/frame... nah, just add one per frame).

Draw order: background, faint pivot anchor, trail, rods, bobs, pivot.

Trail rendering: 
```js
for(i=1;i<trail.length;i++){ a = i/trail.length; ctx.strokeStyle = `hsla(${(trail[i].h)%360},95%,${55+20*a}%,${a*0.9})`; ... }
```
Drawing 700 individual strokes — fine. Could batch in chunks of same alpha but per-segment is simpler and OK.

Also draw lower bob with glow (shadowBlur).

Bob radius ~ 9-11 px scaled: r = Math.max(6, scale*0.07).

Pivot: small circle with cross.

Readout: update text each frame.

Let's write it.

Resize: set canvas width/height with DPR; store layout {cx, cy, scale}.

Now write file.

Also console: no errors expected.

Let me write the complete file.

Positions:
x1 = cx + L1*sin(t1)*scale
y1 = cy + L1*cos(t1)*scale   (θ from downward vertical, y down positive → y = cy + L cosθ) ✓
x2 = x1 + L2*sin(t2)*scale
y2 = y1 + L2*cos(t2)*scale

Good — rods constant length by construction.

Energy sanity check maybe in readout: skip.

Write now.

After writing, use browser: reload, screenshot, wait between screenshots (screenshots don't advance time automatically? Each tool call takes time; motion action exists — "motion" likely captures frames over time). I'll take screenshot at t0, then screenshot again later, then inspect to read readouts.

Let's go.

Let me write the file quickly, then verify in browser.

Double pendulum equations with equal masses m, lengths L, gravity g.

Standard equations for double pendulum (equal masses and lengths):

θ1'' = (-g(2m1+m2)sinθ1 - m2 g sin(θ1-2θ2) - 2 sin(θ1-θ2) m2 (ω2² L2 + ω1² L1 cos(θ1-θ2))) / (L1 (2m1+m2 - m2 cos(2θ1-2θ2)))

θ2'' = (2 sin(θ1-θ2) (ω1² L1 (m1+m2) + g(m1+m2)cosθ1 + ω2² L2 m2 cos(θ1-θ2))) / (L2 (2m1+m2 - m2 cos(2θ1-2θ2)))

Use RK4 with dt = 0.002 (substeps), rendering at fixed step accumulator.

Trail: array of lower bob positions, capped length, drawn with fading alpha and hue over index.

Angles measured from downward vertical: θ=0 is down. Initial θ1=120°, θ2=-10°.

Position: pivot at (cx, cy). x1 = cx + L sinθ1, y1 = cy + L cosθ1 (y down positive, gravity along +y). Actually with downward vertical as reference and y screen down, gravity pulls in +y, so equation of motion uses -g sinθ for θ measured... Let's check: with y down, angle θ from downward vertical, position (L sinθ, L cosθ). Potential energy U = -m g L cosθ (lower is larger cos → smaller U... y = L cosθ, U = m g y = m g L cosθ in screen coords where down is lower... physically U = -m g L cosθ with upward positive). Anyway standard double pendulum EOM as above assumes θ=0 is hanging down, gravity restoring. That's what we want.

Write file.

Let me verify in browser quickly.

Energy drift is huge — relative drift 100x. That suggests either timestep too large or equations wrong. Let's check: relative drift 1e2 means energy grew from 0.149 to 15.1. That's a big instability. Hmm, but state remains finite.

Wait, maybe my equations are wrong. Let me re-derive standard double pendulum equations for equal masses.

Standard (Wikipedia) for m1, m2, L1, L2:

θ1'' = (−g(2m1+m2)sinθ1 − m2 g sin(θ1−2θ2) − 2 sin(θ1−θ2) m2 (ω2² L2 + ω1² L1 cos(θ1−θ2))) / (L1 (2m1+m2 − m2 cos(2θ1−2θ2)))

θ2'' = (2 sin(θ1−θ2) (ω1² L1 (m1+m2) + g(m1+m2) cosθ1 + ω2² L2 m2 cos(θ1−θ2))) / (L2 (2m1+m2 − m2 cos(2θ1−2θ2)))

With m1=m2=M: denominator factor 2M+M − M cos(2δ) = M(3 − cos 2δ). My code used denom = 2M + M cos(2δ)... wrong sign! I wrote `2*M + M*Math.cos(2*d)` but should be `2*M - M*Math.cos(2*d)`. Also in a1 numerator: -G*(2M+M) sin th1 = -3GM sin th1 ✓; - M G sin(th1-2th2) ✓; -2 sinδ M (ω2² L2 + ω1² L1 cosδ) ✓. Divide by L1 * denom ✓.

a2 numerator: 2 sinδ (ω1² L1 (2M) + g(2M) cosθ1 + ω2² L M cosδ) ✓ divided by L2*denom ✓.

So the only error is the sign in denom: should be `2*M - M*Math.cos(2*delta)`. Let me fix in index.html and re-test.