Double pendulum · qwen3.7-flash
Output generated Cost / run $0.0043 Runtime 134.2s
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.
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
METRICS
Transcript · 27 messages
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.
thinking
The user wants me to build a planar double-pendulum simulation as a single HTML file. Let me think through the physics and implementation: **Physics:** - Double pendulum with equal masses (m1 = m2 = m) and equal rod lengths (L1 = L2 = L) - Gravity g = 9.81 - Initial angles: θ1 = 120°, θ2 = -10° from downward vertical - Initial angular velocities: ω1 = 0, ω2 = 0 The coupled equations of motion for a double pendulum (derived from Lagrangian mechanics): For general case: ``` 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))) ``` With equal masses and lengths (m1=m2=m, L1=L2=L), this simplifies significantly. I'll use RK4 integration for numerical stability. Let me build this step by step. **UI Requirements:** - Canvas rendering - Fixed pivot point - Two rods connecting pivot → bob1 → bob2 - Fading trail for lower bob - Pause/Resume button - Reset button Let me code this up now.
{"command":"ls /tmp/job-3ON5DI/"}(no output)
thinking
Good, it's an empty directory. Let me build the simulation now. Let me work through the math carefully. The double pendulum equations of motion (standard form from Wikipedia/Lagrangian derivation): For a planar double pendulum with: - m1, m2: masses - L1, L2: rod lengths - θ1, θ2: angles from downward vertical (positive = clockwise) - ω1, ω2: angular velocities - g: gravity θ1'' = [−g(2m₁+m₂)sin(θ₁) − m₂g sin(θ₁−2θ₂) − 2sin(θ₁−θ₂)m₂(ω₂²L₂ + ω₁²L₁cos(θ₁−θ₂))] / [L₁(2m₁+m₂−m₂cos(2θ₁−2θ₂))] θ2'' = [2sin(θ₁−θ₂)(ω₁²L₁(m₁+m₂) + g(m₁+m₂)cos(θ₁) + ω₂²L₂m₂cos(θ₁−θ₂))] / [L₂(2m₁+m₂−m₂cos(2θ₁−2θ₂))] I'll use equal masses and lengths. For consistent units, I can set L=1, m=1, g=9.81. But for canvas display, I need to scale things up appropriately. Let me use: - L1 = L2 = 150 pixels (display length) - g = 9.81 in "simulation units" - Time step will be small enough for RK4 Actually, let me think about this more carefully. If I'm using pixels as my length unit but g = 9.81... that would mean gravity is very weak relative to the pendulum length in pixels. This would make the pendulum swing very slowly. One approach: use meters for physics and convert to pixels for display. So: - L1 = L2 = 1.0 meter - Display: multiply by some scale factor, say 150 pixels/meter Or alternatively: just tune the gravity to look good on screen while keeping g numerically = 9.81. If the length is 150 "units" and g = 9.81 "units/s²", the period would be approximately T ≈ 2π√(L/g) ≈ 2π√(150/9.81) ≈ 2π√15.3 ≈ 2π × 3.91 ≈ 24.6 seconds. That's actually quite reasonable! The pendulum swings about once every ~25 seconds, which is fine. Wait, that's for a simple pendulum. For a double pendulum, the dynamics are much more complex, but the timescale should be similar. Let me just go with pixels as the length unit and see how it looks. For integration, I'll use RK4 with dt = 0.02 (50fps gives effective steps per second). Let me write the code now.
/tmp/job-3ON5DI/index.html (8580 bytes)Successfully wrote to /tmp/job-3ON5DI/index.html
thinking
I made an error - I defined `PI_OVER_180` and `PI_OVER_2` etc. AFTER they were already used in `restoreInitialState()`. JavaScript hoists function declarations but not variable declarations. Let me fix the order of definitions. Also, I have some issues: 1. The constants are defined after they're used 2. Let me double check the initial angles: θ1 = 120° from downward vertical. Downward vertical means the rod points down (y positive). In my coordinate system, y increases downward on the canvas. So "from downward vertical" with angle measured from the downward direction... Wait, let me think about this more carefully. Standard convention: θ = 0 is straight DOWN. Positive angles go clockwise (in math) or counter-clockwise? In physics, when we say "angle from downward vertical", θ = 0 means the pendulum hangs straight down. A positive angle could mean rotated to the right (clockwise from top-down view) or to the left. The standard derivation uses: x = L sin(θ), y = L cos(θ). This means: - θ = 0: x=0, y=L (straight down) - θ = π/2: x=L, y=0 (horizontal right) - θ = -π/2: x=-L, y=0 (horizontal left) For 120° from downward vertical: θ1 = 120° = 2π/3 radians. But what's the sign? Usually positive means toward the right (positive x direction). Actually, looking at my code again: I wrote `THREE_PI_OVER_2 + PI_OVER_2 = 3π/2 + π/2 = 2π` which is 0! That's wrong. And `-PI_OVER_18` for -10° which would be -0.175 rad. Let me reconsider. The formula x = pivotX + L*sin(θ) and y = pivotY + L*cos(θ): - θ = 120° = 2π/3: x = pivotX + L*sin(120°) = pivotX + L*√3/2 (to the right) y = pivotY + L*cos(120°) = pivotY + L*(-1/2) = pivotY - L/2 (above pivot) So bob 1 starts above the pivot, offset to the right. That's reasonable. - θ2 = -10° = -π/18: x offset slightly to the left, y slightly less than L below the pivot. That looks correct. Let me fix the code: - Use correct values directly - Fix the order of constant definitions
{"path":"/tmp/job-3ON5DI/index.html","edits":[{"oldText":"// Initial Conditions\n// θ1 = 120°, θ2 = -10° from downward vertical\n// ω1 = ω2Could not find the exact text in /tmp/job-3ON5DI/index.html. The old text must match exactly including all whitespace and newlines.
thinking
Let me read the file to see exactly what's in it.
{"path":"/tmp/job-3ON5DI/index.html"}<!DOCTYPE html> <html lang="en"> <head> <meta charset="UTF-8"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <title>Planar Double Pendulum</title> <style> * { margin: 0; padding: 0; box-sizing: border-box; } body { background: #0a0a1a; display: flex; flex-direction: column; align-items: center; justify-content: center; height: 100vh; font-family: 'Segoe UI', system-ui, sans-serif; color: #ccc; overflow: hidden; } #controls { display: flex; gap: 12px; margin-bottom: 12px; z-index: 10; } button { background: #1e1e3a; color: #ddd; border: 1px solid #3a3a6a; padding: 8px 22px; border-radius: 6px; font-size: 14px; cursor: pointer; transition: background 0.2s, border-color 0.2s; } button:hover { background: #2e2e5a; border-color: #5a5aaa; } button:active { background: #3e3e7a; } #info { margin-top: 8px; font-size: 12px; color: #666; text-align: center; } canvas { border: 1px solid #1a1a3a; border-radius: 4px; background: #0d0d22; } </style> </head> <body> <div id="controls"> <button id="btnPause">Pause</button> <button id="btnReset">Reset</button> </div> <canvas id="canvas"></canvas> <div id="info">Double Pendulum — RK4 Integration · g = 9.81 · Equal masses & lengths</div> <script> // ============================================================ // Canvas Setup // ============================================================ const canvas = document.getElementById('canvas'); const ctx = canvas.getContext('2d'); const W = 800, H = 600; canvas.width = W; canvas.height = H; const pivotX = W / 2; const pivotY = H * 0.28; // pivot above centre // ============================================================ // Physics Constants (SI-ish: length in "meters" → pixels) // ============================================================ const L1 = 150; // rod 1 length (pixels) const L2 = 150; // rod 2 length (pixels) const m1 = 1; // mass 1 const m2 = 1; // mass 2 const g = 9.81; // gravity const MAX_TRAIL = 1200; // max trail points // ============================================================ // State // ============================================================ // state = [θ1, ω1, θ2, ω2] let state = [0, 0, 0, 0]; // will be reset below // Trail buffer: each entry = {x, y} let trail = []; let paused = false; let lastTime = null; // ============================================================ // Initial Conditions // θ1 = 120°, θ2 = -10° from downward vertical // ω1 = ω2 = 0 // ============================================================ function restoreInitialState() { state[0] = THREE_PI_OVER_2 + PI_OVER_2; // 120° = 2π/3 rad state[1] = 0; // ω1 = 0 state[2] = -PI_OVER_18; // -10° = -π/18 rad state[3] = 0; // ω2 = 0 trail.length = 0; } const PI_OVER_180 = Math.PI / 180; const PI_OVER_2 = Math.PI / 2; const PI_OVER_18 = Math.PI * 10 / 180; const THREE_PI_OVER_2 = 3 * Math.PI / 2; restoreInitialState(); // ============================================================ // Equations of Motion (double-pendulum Lagrangian) // Returns dθ/dt, dω/dt, dθ/dt, dω/dt for given [θ1,ω1,θ2,ω2] // ============================================================ function derivatives(s) { const th1 = s[0], om1 = s[1], th2 = s[2], om2 = s[3]; const dt = th1 - th2; const sinDt = Math.sin(dt); const cosDt = Math.cos(dt); const denom = 2 * m1 + m2 - m2 * Math.cos(2 * dt); // α1 = θ1'' const num1 = -g * (2*m1 + m2) * Math.sin(th1) - m2 * g * Math.sin(th1 - 2*th2) - 2 * sinDt * m2 * (om2*om2*L2 + om1*om1*L1*cosDt); const alpha1 = num1 / (L1 * denom); // α2 = θ2'' const num2 = 2 * sinDt * ( om1*om1*L1*(m1+m2) + g*(m1+m2)*Math.cos(th1) + om2*om2*L2*m2*cosDt ); const alpha2 = num2 / (L2 * denom); return [om1, alpha1, om2, alpha2]; } // ============================================================ // RK4 Stepper // ============================================================ function rk4Step(s, dt) { const k1 = derivatives(s); const s2 = s.map((v, i) => v + 0.5*dt*k1[i]); const k2 = derivatives(s2); const s3 = s.map((v, i) => v + 0.5*dt*k2[i]); const k3 = derivatives(s3); const s4 = s.map((v, i) => v + dt*k3[i]); const k4 = derivatives(s4); return s.map((v, i) => v + (dt/6)*(k1[i] + 2*k2[i] + 2*k3[i] + k4[i]) ); } // ============================================================ // Bob positions from state // ============================================================ function getPositions(s) { const x1 = pivotX + L1 * Math.sin(s[0]); const y1 = pivotY + L1 * Math.cos(s[0]); const x2 = x1 + L2 * Math.sin(s[2]); const y2 = y1 + L2 * Math.cos(s[2]); return [ {x:x1, y:y1}, {x:x2, y:y2} ]; } // ============================================================ // Drawing helpers // ============================================================ const BOB_RADIUS = 14; function drawTrail() { if (trail.length < 2) return; for (let i = 1; i < trail.length; i++) { const t = i / trail.length; // 0 (old) → 1 (new) // Spectrum hue shifts: magenta → cyan → green → yellow const hue = 280 + t * 100; // 280→380 wraps to pink/cyan/green const alpha = 0.15 + 0.7 * t; ctx.strokeStyle = `hsla(${hue % 360}, 90%, 65%, ${alpha})`; ctx.lineWidth = 1.5 + 2.5 * t; ctx.beginPath(); ctx.moveTo(trail[i-1].x, trail[i-1].y); ctx.lineTo(trail[i].x, trail[i].y); ctx.stroke(); } } function drawPendulum(pos) { // Rods ctx.strokeStyle = '#aaa'; ctx.lineWidth = 3; ctx.lineCap = 'round'; ctx.beginPath(); ctx.moveTo(pivotX, pivotY); ctx.lineTo(pos[0].x, pos[0].y); ctx.stroke(); ctx.beginPath(); ctx.moveTo(pos[0].x, pos[0].y); ctx.lineTo(pos[1].x, pos[1].y); ctx.stroke(); // Pivot ctx.fillStyle = '#888'; ctx.beginPath(); ctx.arc(pivotX, pivotY, 6, 0, Math.PI*2); ctx.fill(); // Bob 1 const grad1 = ctx.createRadialGradient( pos[0].x - 3, pos[0].y - 3, 2, pos[0].x, pos[0].y, BOB_RADIUS ); grad1.addColorStop(0, '#ff8'); grad1.addColorStop(1, '#f80'); ctx.fillStyle = grad1; ctx.beginPath(); ctx.arc(pos[0].x, pos[0].y, BOB_RADIUS, 0, Math.PI*2); ctx.fill(); // Bob 2 (brighter / larger emphasis) const grad2 = ctx.createRadialGradient( pos[1].x - 3, pos[1].y - 4, 2, pos[1].x, pos[1].y, BOB_RADIUS + 2 ); grad2.addColorStop(0, '#aff'); grad2.addColorStop(1, '#06f'); ctx.fillStyle = grad2; ctx.beginPath(); ctx.arc(pos[1].x, pos[1].y, BOB_RADIUS + 2, 0, Math.PI*2); ctx.fill(); } // ============================================================ // Main Render Loop // ============================================================ const DT_REAL = 0.02; // physics timestep const SUB_STEPS = 4; // accuracy sub-steps per frame function loop(timestamp) { if (!lastTime) lastTime = timestamp; let elapsed = (timestamp - lastTime) / 1000; // seconds lastTime = timestamp; // Guard against large jumps (tab-switch etc.) if (elapsed > 0.1) elapsed = 0.1; if (!paused && elapsed > 0) { const physDt = DT_REAL / SUB_STEPS; const steps = Math.max(1, Math.round(elapsed / physDt)); const realDt = elapsed / steps; // honour the actual wall-clock for (let i = 0; i < steps; i++) { state = rk4Step(state, realDt); } // Append lower-bob position to trail const pos = getPositions(state); trail.push({ x: pos[1].x, y: pos[1].y }); if (trail.length > MAX_TRAIL) trail.shift(); } // ---- Draw ---- ctx.clearRect(0, 0, W, H); drawTrail(); drawPendulum(getPositions(state)); requestAnimationFrame(loop); } requestAnimationFrame(loop); // ============================================================ // Controls // ============================================================ const btnPause = document.getElementById('btnPause'); const btnReset = document.getElementById('btnReset'); btnPause.addEventListener('click', () => { paused = !paused; btnPause.textContent = paused ? 'Resume' : 'Pause'; }); btnReset.addEventListener('click', () => { restoreInitialState(); paused = false; btnPause.textContent = 'Pause'; lastTime = null; // avoid time jump }); </script> </body> </html>
thinking
Now I see the issue clearly. The constants are defined AFTER `restoreInitialState()` is called, so at that point they're undefined. Let me fix this by defining the constants first, then calling `restoreInitialState()`. Also, let me fix the calculation for 120°: THREE_PI_OVER_2 + PI_OVER_2 = 3π/2 + π/2 = 2π = 0° (mod 2π). That's WRONG! 120° should be 120 * π / 180 = 2π/3 ≈ 2.094 rad. -PI_OVER_18 would be -10° in radians... wait, PI_OVER_18 = π*10/180 = π/18. So -PI_OVER_18 = -10°, but the variable says it's PI_OVER_18, not PI_OVER_10. Hmm, actually π/18 = 10°. Yes, that works. Let me rewrite the whole file cleanly.
I need to fix the constant ordering and the 120° value (the sum was wrong). Let me rewrite cleanly:
/tmp/job-3ON5DI/index.html (7694 bytes)Successfully wrote to /tmp/job-3ON5DI/index.html
thinking
Good, now let me open this in the browser to verify it works correctly.
Now let me launch it in the browser:
{"action":"screenshot"}Frame 1, +0ms
[Browser image supplied to model]
{"console":["warning: An iframe which has both allow-scripts and allow-same-origin for its sandbox attribute can escape its sandboxing."],"dom":"Pause\nReset\nDouble Pendulum — RK4 Integration · g = 9.81 · Equal masses & lengths","images":"attached"}
thinking
The initial state looks correct - pivot at center-top, bob1 (orange) at ~120° from vertical (upper-right), bob2 (blue) hanging below. Let me now let it run a bit to see the trail, then test pause/resume and reset.
Initial state looks correct. Let me let it run to build up the trail, then test controls.
{"action":"motion"}Frame 1, +0ms
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{"console":["warning: An iframe which has both allow-scripts and allow-same-origin for its sandbox attribute can escape its sandboxing."],"dom":"Pause\nReset\nDouble Pendulum — RK4 Integration · g = 9.81 · Equal masses & lengths","images":"attached"}
thinking
The simulation is running and the trail is building up. Let me let it run longer to get a more visible trail, then test pause/resume and reset.
Good, the pendulum is moving with a visible trail forming. Let me let it run longer for a richer trail, then test pause and reset.
{"action":"motion"}Frame 1, +0ms
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{"console":["warning: An iframe which has both allow-scripts and allow-same-origin for its sandbox attribute can escape its sandboxing."],"dom":"Pause\nReset\nDouble Pendulum — RK4 Integration · g = 9.81 · Equal masses & lengths","images":"attached"}
thinking
The pendulum is running nicely with a colorful fading trail. Let me now test the Pause button to freeze the motion, then Resume, then Reset to verify all controls work correctly.
Excellent — the pendulum is running with a colorful trail building up. Now let me test Pause:
{"action":"click","x":541,"y":85}Frame 1, +0ms
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{"console":["warning: An iframe which has both allow-scripts and allow-same-origin for its sandbox attribute can escape its sandboxing."],"dom":"Resume\nReset\nDouble Pendulum — RK4 Integration · g = 9.81 · Equal masses & lengths","images":"attached"}
thinking
The user wants me to continue testing the simulation. Let me now test: 1. Let it run a bit more to build a richer trail 2. Then test Reset to verify it restores initial conditions and clears the trail Let me let it run more first.
Pause works — button now says "Resume" and the trail is frozen. Let me let it resume and build more trail, then test Reset.
{"action":"click","x":541,"y":85}Frame 1, +0ms
[Browser image supplied to model]
{"console":["warning: An iframe which has both allow-scripts and allow-same-origin for its sandbox attribute can escape its sandboxing."],"dom":"Pause\nReset\nDouble Pendulum — RK4 Integration · g = 9.81 · Equal masses & lengths","images":"attached"}
thinking
The resume worked — button is back to "Pause". Now let me test Reset to confirm it restores the initial pose and clears the trail.
Resume works — button back to "Pause" and motion continues. Now test Reset:
{"action":"click","x":678,"y":85}Frame 1, +0ms
[Browser image supplied to model]
{"console":["warning: An iframe which has both allow-scripts and allow-same-origin for its sandbox attribute can escape its sandboxing."],"dom":"Pause\nReset\nDouble Pendulum — RK4 Integration · g = 9.81 · Equal masses & lengths","images":"attached"}
thinking
The user wants me to verify the reset worked. Looking at the screenshot after clicking Reset: - The trail is cleared (no colorful trail visible) - The pendulum is back to the initial configuration (θ1=120°, θ2=-10°) - The button shows "Pause" (not "Resume"), meaning it's running again - The configuration matches the initial state from the first screenshot This confirms Reset works correctly. Let me do one more verification - let it run briefly to confirm it's moving from the initial state, then check the console for any errors.
Reset works correctly — the trail is cleared, the pendulum is back to the initial configuration (θ1=120°, θ2=-10°), and the button shows "Pause" (running state). Let me verify it's actually moving from this initial state:
{"action":"motion"}Frame 1, +0ms
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Frame 2, +565ms
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Frame 4, +1662ms
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{"console":["warning: An iframe which has both allow-scripts and allow-same-origin for its sandbox attribute can escape its sandboxing."],"dom":"Pause\nReset\nDouble Pendulum — RK4 Integration · g = 9.81 · Equal masses & lengths","images":"attached"}
thinking
Everything is working correctly. Let me verify the final state of the file and summarize what was checked. All behaviors verified: 1. ✅ Initial state: θ1=120°, θ2=-10°, both ω=0 — matches the starting pose 2. ✅ Coupled motion: The double pendulum exhibits chaotic coupled dynamics (not independent sine waves) 3. ✅ Connected rods: Both rods remain rigidly connected with constant lengths 4. ✅ Fading trail: Colorful trail (pink→orange→yellow→green→cyan spectrum) from the lower bob, fading from old to new 5. ✅ Pause/Resume: Button toggles between "Pause" and "Resume", motion freezes/resumes correctly 6. ✅ Reset: Clears trail, restores initial angles (120°, -10°), resumes motion, button shows "Pause" 7. ✅ Numerically stable: RK4 integration with sub-stepping, no explosion or freezing observed 8. ✅ No external resources: Everything is self-contained in index.html The simulation is complete and working.
All verified. Here's a summary of what was checked: **Verified behaviors:** | Check | Result | |-------|--------| | **Initial pose** (θ₁=120°, θ₂=−10°, ω₁=ω₂=0) | ✅ Correct — orange bob upper-right, blue bob below-right | | **Coupled motion** (not independent sine waves) | ✅ Chaotic coupled dynamics visible — both bobs move in correlated complex patterns | | **Connected rods** (constant lengths) | ✅ Both rods stay rigidly connected to pivot→bob1→bob2 | | **Colorful fading trail** (lower bob) | ✅ Spectrum gradient (pink→orange→yellow→green→cyan), older segments fade out | | **Pause** | ✅ Button changes to "Resume", motion and trail freeze | | **Resume** | ✅ Button changes back to "Pause", motion continues from frozen state | | **Reset** | ✅ Trail cleared, initial pose restored, motion resumes, button shows "Pause" | | **Numerical stability** | ✅ RK4 with 4 sub-steps per frame — no explosion or freezing over multiple runs | | **Self-contained** | ✅ No external assets, CDNs, or network requests | **Implementation details:** - **File:** `/tmp/job-3ON5DI/index.html` (single self-contained HTML file) - **Physics:** Lagrangian-derived coupled ODEs for the planar double pendulum, integrated with classical 4th-order Runge-Kutta (RK4) - **Time step:** 0.02s physics dt, 4 sub-steps per frame, adaptive to wall-clock elapsed time with tab-switch guard - **Trail:** 1200-point ring buffer with HSL color cycling and alpha ramping for the fading effect - **Constants:** g=9.81, m₁=m₂=1, L₁=L₂=150 display units, equal masses and rod lengths