Double pendulum · nemotron-3-super-120b-a12b

We need to create index.html with a double pendulum simulation. We'll write the HTML, CSS, JS inline.

We need to implement physics: double pendulum with equal masses m, equal lengths L, gravity g=9.81. Initial angles: theta1 = 120 degrees (from downward vertical? Usually angle from vertical down: 0 is hanging down. They said measured from downward vertical. So downward vertical is 0 degrees? Actually "measured from downward vertical" means angle=0 when pointing straight down. So initial angles: theta1 = 120°, theta2 = -10°. Zero initial angular velocities.

We need to simulate using equations of motion. We can use Lagrangian derivation or use standard formulas. We'll implement using numerical integration (e.g., Runge-Kutta 4) for stability.

We'll have variables: theta1, theta2, omega1, omega2 (angular velocities). Derivatives: dtheta1/dt = omega1, dtheta2/dt = omega2. Then we need alpha1, alpha2 (angular accelerations). We can compute using equations:

For double pendulum with equal masses m and equal lengths L:

Let:
delta = theta2 - theta1
den1 = (m1 + m2) * L - m2 * L * cos(delta)*cos(delta) ??? Actually standard formulas:

Using Lagrangian:

T = 0.5 * m1 * (L1 * dot(theta1))^2 + 0.5 * m2 * [ (L1*dot(theta1))^2 + (L2*dot(theta2))^2 + 2 * L1 * L2 * dot(theta1)*dot(theta2)*cos(theta1 - theta2) ]
V = -m1 * g * L1 * cos(theta1) - m2 * g * (L1*cos(theta1) + L2*cos(theta2))

But with equal masses m, equal lengths L, we can simplify.

Alternatively we can use known equations:

alpha1 = ( -g * (2*m1 + m2) * sin(theta1) - m2 * g * sin(theta1 - 2*theta2) - 2 * sin(theta1 - theta2) * m2 * (omega2^2 * L2 + omega1^2 * L1 * cos(theta1 - theta2)) ) / ( L1 * (2*m1 + m2 - m2 * cos(2*theta1 - 2*theta2)) )
alpha2 = ( 2 * sin(theta1 - theta2) * ( omega1^2 * L1 * (m1 + m2) + g * (m1 + m2) * cos(theta1) + omega2^2 * L2 * m2 * cos(theta1 - theta2) ) ) / ( L2 * (2*m1 + m2 - m2 * cos(2*theta1 - 2*theta2)) )

But that's messy. Let's derive using Lagrangian with m1=m2=m, L1=L2=L.

Let’s compute using sympy? Not possible. We'll implement using standard formulas from Wikipedia double pendulum.

From Wikipedia (double pendulum) for equal masses and equal lengths:

Let:
c = cos(theta1 - theta2)
s = sin(theta1 - theta2)

Denominator: L * (2*m - m*c*c) ??? Actually:

From Wikipedia:

For equal masses m and equal lengths L:

α1 = [ -g*(2*sin(theta1) - sin(theta2)*c) - (omega2^2*L*s + omega1^2*L*s*c)*s ] / [ L*(3 - c*c) ]
α2 = [ 2*c*( g*sin(theta1) + omega1^2*L*s ) + omega2^2*L*s ] / [ L*(3 - c*c) ]

Wait I'm not sure. Let's search memory: I recall formulas:

Let:
num1 = -g*(2*m1 + m2)*sin(theta1) - m2*g*sin(theta1 - 2*theta2) - 2*s* m2 * (omega2^2*L2 + omega1^2*L1*c)
den = L1*(2*m1 + m2 - m2*c*c)

alpha1 = num1 / den

num2 = 2*s * ( omega1^2*L1*(m1+m2) + g*(m1+m2)*cos(theta1) + omega2^2*L2*m2*c )
alpha2 = num2 / (L2*(2*m1 + m2 - m2*c*c))

But with m1=m2=m, L1=L2=L:

den = L*(2m + m - m*c*c) = L*(3m - m*c*c) = m*L*(3 - c*c)

num1 = -g*(2m+m)*sin(theta1) - m*g*sin(theta1 - 2*theta2) - 2*s*m*(omega2^2*L + omega1^2*L*c)
= -g*3m*sin(theta1) - m*g*sin(theta1 - 2*theta2) - 2*m*L*s*(omega2^2 + omega1^2*c)

alpha1 = [ -3g*sin(theta1) - g*sin(theta1 - 2*theta2) - 2*L*s*(omega2^2 + omega1^2*c) ] / [ L*(3 - c*c) ]

num2 = 2*s * ( omega1^2*L*(2m) + g*2m*cos(theta1) + omega2^2*L*m*c )
= 2*s * ( 2m*L*omega1^2 + 2m*g*cos(theta1) + m*L*omega2^2*c )
= 2*m*s * ( 2L*omega1^2 + 2g*cos(theta1) + L*omega2^2*c )

alpha2 = [ 2*m*s * (2L*omega1^2 + 2g*cos(theta1) + L*omega2^2*c) ] / [ L*(3 - c*c) * m ]? Wait denominator: L2*(2*m1 + m2 - m2*c*c) = L*(2m+m - m*c*c) = L*(3m - m*c*c) = m*L*(3 - c*c)

So alpha2 = [2*m*s*(2L*omega1^2 + 2g*cos(theta1) + L*omega2^2*c)] / [m*L*(3 - c*c)] = [2*s*(2L*omega1^2 + 2g*cos(theta1) + L*omega2^2*c)] / [L*(3 - c*c)]

Simplify: alpha2 = [2*s*(2L*omega1^2 + 2g*cos(theta1) + L*omega2^2*c)] / [L*(3 - c*c)]

We can factor L: numerator: 2*s*L*(2*omega1^2 + (2g/L)*cos(theta1) + omega2^2*c). Denominator: L*(3 - c*c). Cancel L:

alpha2 = [2*s*(2*omega1^2 + (2g/L)*cos(theta1) + omega2^2*c)] / (3 - c*c)

Similarly alpha1: numerator: -3g*sin(theta1) - g*sin(theta1 - 2*theta2) - 2*L*s*(omega2^2 + omega1^2*c). Denominator: L*(3 - c*c). So:

alpha1 = [ -3g*sin(theta1) - g*sin(theta1 - 2*theta2) - 2*L*s*(omega2^2 + omega1^2*c) ] / [ L*(3 - c*c) ]

We can write as:

alpha1 = ( -3*(g/L)*sin(theta1) - (g/L)*sin(theta1 - 2*theta2) - 2*s*(omega2^2 + omega1^2*c) ) / (3 - c*c)

Because dividing numerator and denominator by L: numerator becomes -3g*sin(theta1)/L - g*sin(theta1-2*theta2)/L - 2*s*(omega2^2 + omega1^2*c). Denominator becomes (3 - c*c). So:

alpha1 = ( -3*(g/L)*sin(theta1) - (g/L)*sin(theta1 - 2*theta2) - 2*s*(omega2^2 + omega1^2*c) ) / (3 - c*c)

alpha2 = ( 2*s*(2*omega1^2 + (2g/L)*cos(theta1) + omega2^2*c) ) / (3 - c*c)

Where s = sin(theta1 - theta2), c = cos(theta1 - theta2).

Check: This seems plausible.

We'll implement using these equations.

We'll use RK4 integration.

State vector: [theta1, theta2, omega1, omega2]

Derivatives: dtheta1/dt = omega1, dtheta2/dt = omega2, domega1/dt = alpha1, domega2/dt = alpha2.

We'll set g = 9.81, L = 1.0 (we can choose length scaling; we need to draw on canvas, so we can set L = 100 pixels maybe). Actually we need to keep rod lengths constant and visible. We'll set L = 100 (so each rod length 100px). Then we need to scale drawing accordingly. We'll set pivot at center x = width/2, y = some offset (maybe 150). Actually we can set pivot at (width/2, 100). Then each rod length L = 120 maybe.

We'll decide later.

We'll need to clear trail: we'll store an array of points for lower bob positions, with limited length (e.g., 500 points). Each frame we add current lower bob position, and when drawing we draw lines between points with fading alpha based on age.

We'll implement pause/resume and reset buttons.

Now we need to write index.html.

Let's start by creating the file.

First, check current directory: we are at /tmp/job-3fOU5T.

We'll write index.html.

We'll use bash to write file.

Let's create a skeleton.

We'll do:

<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<title>Double Pendulum Simulation</title>
<style>
  /* CSS */
</style>
</head>
<body>
  <canvas id="canvas"></canvas>
  <div id="controls">
    <button id="pauseResume">Pause</button>
    <button id="reset">Reset</button>
  </div>
<script>
  // JS
</script>
</body>
</html>

We'll then edit with actual content.

Let's start by writing the file.

We'll use write command.

Now we need to edit the file to add the JavaScript and maybe adjust CSS.

We'll edit using edit tool. Let's first read the file to see its content.

Now we need to replace the script content with actual simulation.

We'll create a script that sets up canvas, variables, physics, animation loop, and event listeners.

We'll write a new script block.

We'll use edit to replace the line "// Simulation code will go here" with the actual code. But we need to match exact text. Let's replace the entire script tag content.

We can replace from "<script>" to "</script>" but we need to keep the tags. Simpler: replace the inner content.

We'll do:

<edit path="/tmp/job-3fOU5T/index.html" edits=[{oldText: "  <script>\n  // Simulation code will go here\n</script>", newText: "  <script>\n/* simulation code */\n</script>"}]]

But we need to include the new code.

Let's construct the JavaScript.

We'll write:

const canvas = document.getElementById('canvas');
const ctx = canvas.getContext('2d');

// Set canvas size to window inner size minus some margin, but we'll set fixed size for simplicity.
function resizeCanvas() {
  canvas.width = Math.min(window.innerWidth - 40, 800);
  canvas.height = Math.min(window.innerHeight - 200, 600);
}
window.addEventListener('resize', resizeCanvas);
resizeCanvas();

// Physics constants
const g = 9.81;
const L = 100; // rod length in pixels
const m = 1.0; // mass (arbitrary unit)
const dt = 0.02; // time step in seconds

// State variables
let theta1 = 120 * Math.PI / 180; // initial angle from downward vertical (radians)
let theta2 = -10 * Math.PI / 180;
let omega1 = 0;
let omega2 = 0;

// Trail array for lower bob
const trail = [];
const maxTrailLength = 500;

// Pivot point (center top-ish)
const pivotX = canvas.width / 2;
const pivotY = 150; // adjust as needed

// Simulation control
let paused = false;
let lastTime = 0;

// Compute accelerations
function computeAccelerations() {
  const c = Math.cos(theta1 - theta2);
  const s = Math.sin(theta1 - theta2);
  const denom = 3 - c * c; // denominator common

  const gOverL = g / L;

  const alpha1 = (-3 * gOverL * Math.sin(theta1) - gOverL * Math.sin(theta1 - 2 * theta2) - 2 * s * (omega2 * omega2 + omega1 * omega1 * c)) / denom;
  const alpha2 = (2 * s * (2 * omega1 * omega1 + 2 * gOverL * Math.cos(theta1) + omega2 * omega2 * c)) / denom;

  return {alpha1, alpha2};
}

// RK4 integration
function updateState() {
  const k1 = computeDerivatives(theta1, theta2, omega1, omega2);
  const k2 = computeDerivatives(theta1 + 0.5 * dt * k1.omega1, theta2 + 0.5 * dt * k1.omega2,
                                omega1 + 0.5 * dt * k1.alpha1, omega2 + 0.5 * dt * k1.alpha2);
  const k3 = computeDerivatives(theta1 + 0.5 * dt * k2.omega1, theta2 + 0.5 * dt * k2.omega2,
                                omega1 + 0.5 * dt * k2.alpha1, omega2 + 0.5 * dt * k2.alpha2);
  const k4 = computeDerivatives(theta1 + dt * k3.omega1, theta2 + dt * k3.omega2,
                                omega1 + dt * k3.alpha1, omega2 + dt * k3.alpha2);

  theta1 += dt * (k1.omega1 + 2 * k2.omega1 + 2 * k3.omega1 + k4.omega1) / 6;
  theta2 += dt * (k1.omega2 + 2 * k2.omega2 + 2 * k3.omega2 + k4.omega2) / 6;
  omega1 += dt * (k1.alpha1 + 2 * k2.alpha1 + 2 * k3.alpha1 + k4.alpha1) / 6;
  omega2 += dt * (k1.alpha2 + 2 * k2.alpha2 + 2 * k3.alpha2 + k4.alpha2) / 6;

  // Keep angles reasonable (optional)
  theta1 = ((theta1 % (2 * Math.PI)) + 2 * Math.PI) % (2 * Math.PI);
  theta2 = ((theta2 % (2 * Math.PI)) + 2 * Math.PI) % (2 * Math.PI);
}

function computeDerivatives(t1, t2, o1, o2) {
  const c = Math.cos(t1 - t2);
  const s = Math.sin(t1 - t2);
  const denom = 3 - c * c;
  const gOverL = g / L;

  const alpha1 = (-3 * gOverL * Math.sin(t1) - gOverL * Math.sin(t1 - 2 * t2) - 2 * s * (o2 * o2 + o1 * o1 * c)) / denom;
  const alpha2 = (2 * s * (2 * o1 * o1 + 2 * gOverL * Math.cos(t1) + o2 * o2 * c)) / denom;

  return {omega1: o1, omega2: o2, alpha1, alpha2};
}

// Drawing function
function draw() {
  ctx.clearRect(0, 0, canvas.width, canvas.height);

  // Draw pivot
  ctx.fillStyle = '#fff';
  ctx.beginPath();
  ctx.arc(pivotX, pivotY, 5, 0, Math.PI * 2);
  ctx.fill();

  // Compute positions
  const x1 = pivotX + L * Math.sin(theta1);
  const y1 = pivotY + L * Math.cos(theta1);
  const x2 = x1 + L * Math.sin(theta2);
  const y2 = y1 + L * Math.cos(theta2);

  // Draw rods
  ctx.strokeStyle = '#fff';
  ctx.lineWidth = 2;
  ctx.beginPath();
  ctx.moveTo(pivotX, pivotY);
  ctx.lineTo(x1, y1);
  ctx.stroke();

  ctx.beginPath();
  ctx.moveTo(x1, y1);
  ctx.lineTo(x2, y2);
  ctx.stroke();

  // Draw bobs
  ctx.fillStyle = '#ff6b6b';
  ctx.beginPath();
  ctx.arc(x1, y1, 12, 0, Math.PI * 2);
  ctx.fill();

  ctx.fillStyle = '#4ecdc4';
  ctx.beginPath();
  ctx.arc(x2, y2, 12, 0, Math.PI * 2);
  ctx.fill();

  // Draw trail for lower bob
  if (trail.length > 1) {
    for (let i = 0; i < trail.length - 1; i++) {
      const alpha = (i + 1) / trail.length; // fade from start to end? Actually we want newer points more opaque.
      // We'll compute age: index i from 0 (oldest) to length-1 (newest)
      const ageRatio = i / (trail.length - 1);
      const trailAlpha = 0.4 * (1 - ageRatio); // older more transparent
      ctx.strokeStyle = `rgba(78, 205, 196, ${trailAlpha})`;
      ctx.lineWidth = 2;
      ctx.beginPath();
      ctx.moveTo(trail[i].x, trail[i].y);
      ctx.lineTo(trail[i + 1].x, trail[i + 1].y);
      ctx.stroke();
    }
  }
}

// Main loop
function loop(timestamp) {
  if (!lastTime) lastTime = timestamp;
  const delta = (timestamp - lastTime) / 1000; // seconds
  lastTime = timestamp;

  if (!paused) {
    // Fixed time step for stability
    const steps = Math.ceil(delta / dt);
    for (let i = 0; i < steps; i++) {
      updateState();
      // Add current lower bob position to trail
      const x2 = pivotX + L * Math.sin(theta1) + L * Math.sin(theta2);
      const y2 = pivotY + L * Math.cos(theta1) + L * Math.cos(theta2);
      trail.push({x: x2, y: y2});
      if (trail.length > maxTrailLength) {
        trail.shift();
      }
    }
  }

  draw();
  requestAnimationFrame(loop);
}

// Event listeners
document.getElementById('pauseResume').addEventListener('click', () => {
  paused = !paused;
  const btn = document.getElementById('pauseResume');
  btn.textContent = paused ? 'Resume' : 'Pause';
  btn.classList.toggle('paused', paused);
});

document.getElementById('reset').addEventListener('click', () => {
  theta1 = 120 * Math.PI / 180;
  theta2 = -10 * Math.PI / 180;
  omega1 = 0;
  omega2 = 0;
  trail.length = 0;
  paused = false;
  document.getElementById('pauseResume').textContent = 'Pause';
  document.getElementById('pauseResume').classList.remove('paused');
  lastTime = 0;
});

// Start animation
requestAnimationFrame(loop);
</script>

But we need to check for any errors.

We'll now replace the script content.

Let's construct the new script content as a string.

We'll replace from "<script>" to "</script>" inclusive? Actually we want to keep the tags. We'll replace the inner content.

We'll do:

oldText: "<script>\n  // Simulation code will go here\n</script>"
newText: "<script>\n/* our code */\n</script>"

Let's write the new script content exactly.

We'll need to escape any backticks? We'll just write the string.

Let's create the new script content.

We'll write:

const canvas = document.getElementById('canvas');
const ctx = canvas.getContext('2d');

// Set canvas size to window inner size minus some margin, but we'll set fixed size for simplicity.
function resizeCanvas() {
  canvas.width = Math.min(window.innerWidth - 40, 800);
  canvas.height = Math.min(window.innerHeight - 200, 600);
}
window.addEventListener('resize', resizeCanvas);
resizeCanvas();

// Physics constants
const g = 9.81;
const L = 100; // rod length in pixels
const m = 1.0; // mass (arbitrary unit)
const dt = 0.02; // time step in seconds

// State variables
let theta1 = 120 * Math.PI / 180; // initial angle from downward vertical (radians)
let theta2 = -10 * Math.PI / 180;
let omega1 = 0;
let omega2 = 0;

// Trail array for lower bob
const trail = [];
const maxTrailLength = 500;

// Pivot point (center top-ish)
const pivotX = canvas.width / 2;
const pivotY = 150; // adjust as needed

// Simulation control
let paused = false;
let lastTime = 0;

// Compute accelerations
function computeAccelerations() {
  const c = Math.cos(theta1 - theta2);
  const s = Math.sin(theta1 - theta2);
  const denom = 3 - c * c; // denominator common

  const gOverL = g / L;

  const alpha1 = (-3 * gOverL * Math.sin(theta1) - gOverL * Math.sin(theta1 - 2 * theta2) - 2 * s * (omega2 * omega2 + omega1 * omega1 * c)) / denom;
  const alpha2 = (2 * s * (2 * omega1 * omega1 + 2 * gOverL * Math.cos(theta1) + omega2 * omega2 * c)) / denom;

  return {alpha1, alpha2};
}

// RK4 integration
function updateState() {
  const k1 = computeDerivatives(theta1, theta2, omega1, omega2);
  const k2 = computeDerivatives(theta1 + 0.5 * dt * k1.omega1, theta2 + 0.5 * dt * k1.omega2,
                                omega1 + 0.5 * dt * k1.alpha1, omega2 + 0.5 * dt * k1.alpha2);
  const k3 = computeDerivatives(theta1 + 0.5 * dt * k2.omega1, theta2 + 0.5 * dt * k2.omega2,
                                omega1 + 0.5 * dt * k2.alpha1, omega2 + 0.5 * dt * k2.alpha2);
  const k4 = computeDerivatives(theta1 + dt * k3.omega1, theta2 + dt * k3.omega2,
                                omega1 + dt * k3.alpha1, omega2 + dt * k3.alpha2);

  theta1 += dt * (k1.omega1 + 2 * k2.omega1 + 2 * k3.omega1 + k4.omega1) / 6;
  theta2 += dt * (k1.omega2 + 2 * k2.omega2 + 2 * k3.omega2 + k4.omega2) / 6;
  omega1 += dt * (k1.alpha1 + 2 * k2.alpha1 + 2 * k3.alpha1 + k4.alpha1) / 6;
  omega2 += dt * (k1.alpha2 + 2 * k2.alpha2 + 2 * k3.alpha2 + k4.alpha2) / 6;

  // Keep angles reasonable (optional)
  theta1 = ((theta1 % (2 * Math.PI)) + 2 * Math.PI) % (2 * Math.PI);
  theta2 = ((theta2 % (2 * Math.PI)) + 2 * Math.PI) % (2 * Math.PI);
}

function computeDerivatives(t1, t2, o1, o2) {
  const c = Math.cos(t1 - t2);
  const s = Math.sin(t1 - t2);
  const denom = 3 - c * c;
  const gOverL = g / L;

  const alpha1 = (-3 * gOverL * Math.sin(t1) - gOverL * Math.sin(t1 - 2 * t2) - 2 * s * (o2 * o2 + o1 * o1 * c)) / denom;
  const alpha2 = (2 * s * (2 * o1 * o1 + 2 * gOverL * Math.cos(t1) + o2 * o2 * c)) / denom;

  return {omega1: o1, omega2: o2, alpha1, alpha2};
}

// Drawing function
function draw() {
  ctx.clearRect(0, 0, canvas.width, canvas.height);

  // Draw pivot
  ctx.fillStyle = '#fff';
  ctx.beginPath();
  ctx.arc(pivotX, pivotY, 5, 0, Math.PI * 2);
  ctx.fill();

  // Compute positions
  const x1 = pivotX + L * Math.sin(theta1);
  const y1 = pivotY + L * Math.cos(theta1);
  const x2 = x1 + L * Math.sin(theta2);
  const y2 = y1 + L * Math.cos(theta2);

  // Draw rods
  ctx.strokeStyle = '#fff';
  ctx.lineWidth = 2;
  ctx.beginPath();
  ctx.moveTo(pivotX, pivotY);
  ctx.lineTo(x1, y1);
  ctx.stroke();

  ctx.beginPath();
  ctx.moveTo(x1, y1);
  ctx.lineTo(x2, y2);
  ctx.stroke();

  // Draw bobs
  ctx.fillStyle = '#ff6b6b';
  ctx.beginPath();
  ctx.arc(x1, y1, 12, 0, Math.PI * 2);
  ctx.fill();

  ctx.fillStyle = '#4ecdc4';
  ctx.beginPath();
  ctx.arc(x2, y2, 12, 0, Math.PI * 2);
  ctx.fill();

  // Draw trail for lower bob
  if (trail.length > 1) {
    for (let i = 0; i < trail.length - 1; i++) {
      const ageRatio = i / (trail.length - 1);
      const trailAlpha = 0.4 * (1 - ageRatio); // older more transparent
      ctx.strokeStyle = `rgba(78, 205, 196, ${trailAlpha})`;
      ctx.lineWidth = 2;
      ctx.beginPath();
      ctx.moveTo(trail[i].x, trail[i].y);
      ctx.lineTo(trail[i + 1].x, trail[i + 1].y);
      ctx.stroke();
    }
  }
}

// Main loop
function loop(timestamp) {
  if (!lastTime) lastTime = timestamp;
  const delta = (timestamp - lastTime) / 1000; // seconds
  lastTime = timestamp;

  if (!paused) {
    // Fixed time step for stability
    const steps = Math.ceil(delta / dt);
    for (let i = 0; i < steps; i++) {
      updateState();
      // Add current lower bob position to trail
      const x2 = pivotX + L * Math.sin(theta1) + L * Math.sin(theta2);
      const y2 = pivotY + L * Math.cos(theta1) + L * Math.cos(theta2);
      trail.push({x: x2, y: y2});
      if (trail.length > maxTrailLength) {
        trail.shift();
      }
    }
  }

  draw();
  requestAnimationFrame(loop);
}

// Event listeners
document.getElementById('pauseResume').addEventListener('click', () => {
  paused = !paused;
  const btn = document.getElementById('pauseResume');
  btn.textContent = paused ? 'Resume' : 'Pause';
  btn.classList.toggle('paused', paused);
});

document.getElementById('reset').addEventListener('click', () => {
  theta1 = 120 * Math.PI / 180;
  theta2 = -10 * Math.PI / 180;
  omega1 = 0;
  omega2 = 0;
  trail.length = 0;
  paused = false;
  document.getElementById('pauseResume').textContent = 'Pause';
  document.getElementById('pauseResume').classList.remove('paused');
  lastTime = 0;
});

// Start animation
requestAnimationFrame(loop);

Now we need to replace.

Let's perform the edit.

Now we need to replace the script content with actual simulation.

We'll write a new script block. We'll replace the entire <script> tag content.

We'll do an edit: replace from "<script>" to "</script>" with new content.

But we need to be careful: there may be other script tags? No.

Let's define new script content.

We'll implement:

- Constants: g = 9.81, L = 100 (rod length in pixels), m = 1 (mass arbitrary)
- State: theta1, theta2, omega1, omega2 (in radians)
- Initial: theta1 = 120° in radians, theta2 = -10° in radians, omega1 = 0, omega2 = 0.
- Trail: array of {x, y, age} or we can store points and alpha based on index.
- We'll set maxTrailLength = 200.
- Animation loop using requestAnimationFrame.
- Time step: we can use fixed dt = 0.02 seconds (50 Hz) and use RK4 to integrate.
- We'll have paused flag.
- Functions: computeDerivatives(state) returns [dtheta1, dtheta2, domega1, domega2].
- RK4 step.
- Drawing: clear canvas, draw pivot, rods, bobs, trail.

Let's craft the script.

We'll write it as a string.

Now we need to edit the file.

We'll use edit with oldText being the script tag content (including the comment). We'll replace from "<script>" to "</script>" inclusive? Actually we want to keep the tags but replace inner content. So we can match "<script>\n  // Simulation code will go here\n</script>" and replace with "<script>\n[new code]\n</script>".

Let's do that.

First, let's write the new script content.

We'll create a variable newScript.

But we need to embed it in the edit command.

We'll do:

edit path=/tmp/job-3fOU5T/index.html edits=[{oldText: "<script>\n  // Simulation code will go here\n</script>", newText: "<script>\n[new code]\n</script>"}]

We need to ensure exact match including newline and spaces.

Let's first capture the exact oldText from the file: we saw lines:

<script>
  // Simulation code will go here
</script>

Note there is a newline after opening tag, then two spaces, then comment, then newline, then closing tag.

Let's copy exactly: "<script>\n  // Simulation code will go here\n</script>"

Now new script content.

We'll write:

<script>
  const canvas = document.getElementById('canvas');
  const ctx = canvas.getContext('2d');
  const pauseResumeBtn = document.getElementById('pauseResume');
  const resetBtn = document.getElementById('reset');

  // Set canvas size to window size minus some margin
  function resizeCanvas() {
    canvas.width = Math.min(window.innerWidth - 40, 800);
    canvas.height = Math.min(window.innerHeight - 200, 600);
  }
  window.addEventListener('resize', resizeCanvas);
  resizeCanvas();

  // Physics constants
  const g = 9.81;
  const L = 100; // rod length in pixels
  const m = 1; // mass (arbitrary, cancels out)

  // State: [theta1, theta2, omega1, omega2] in radians and rad/s
  let state = [0, 0, 0, 0];
  let paused = false;
  let trail = [];
  const maxTrailLength = 200;

  // Initial conditions
  function resetSimulation() {
    state[0] = 120 * Math.PI / 180; // theta1
    state[1] = -10 * Math.PI / 180; // theta2
    state[2] = 0; // omega1
    state[3] = 0; // omega2;
    trail = [];
    paused = false;
    pauseResumeBtn.textContent = 'Pause';
    pauseResumeBtn.classList.remove('paused');
  }

  // Derivatives function
  function derivatives(s) {
    const [theta1, theta2, omega1, omega2] = s;
    const delta = theta2 - theta1;
    const c = Math.cos(delta);
    const sDelta = Math.sin(delta);
    const denom = 3 - c * c; // denominator common

    // Compute alpha1 and alpha2 using derived formulas
    const alpha1 = (-3 * (g/L) * Math.sin(theta1) - (g/L) * Math.sin(theta1 - 2*theta2) - 2 * sDelta * (omega2*omega2 + omega1*omega1 * c)) / denom;
    const alpha2 = (2 * sDelta * (2 * omega1*omega1 + (2*g/L) * Math.cos(theta1) + omega2*omega2 * c)) / denom;

    return [omega1, omega2, alpha1, alpha2];
  }

  // Runge-Kutta 4 step
  function rk4Step(s, dt) {
    const k1 = derivatives(s);
    const k2 = derivatives(s.map((v, i) => v + 0.5 * dt * k1[i]));
    const k3 = derivatives(s.map((v, i) => v + 0.5 * dt * k2[i]));
    const k4 = derivatives(s.map((v, i) => v + dt * k3[i]));
    return s.map((v, i) => v + (dt / 6) * (k1[i] + 2*k2[i] + 2*k3[i] + k4[i]));
  }

  // Drawing function
  function draw() {
    ctx.clearRect(0, 0, canvas.width, canvas.height);

    // Pivot point (center top)
    const pivotX = canvas.width / 2;
    const pivotY = 100; // distance from top

    // Compute positions
    const x1 = pivotX + L * Math.sin(state[0]);
    const y1 = pivotY + L * Math.cos(state[0]);
    const x2 = x1 + L * Math.sin(state[1]);
    const y2 = y1 + L * Math.cos(state[1]);

    // Draw trail of lower bob
    if (trail.length > 1) {
      for (let i = 0; i < trail.length - 1; i++) {
        const alpha = (i + 1) / trail.length; // fade from transparent to opaque? Actually we want older more transparent.
        // Let's make alpha based on age: newer points more opaque.
        const ageAlpha = (i + 1) / trail.length; // i=0 oldest -> low alpha, i=trail.length-1 newest -> high alpha
        ctx.strokeStyle = `rgba(255, 100, 100, ${ageAlpha * 0.6})`; // reddish trail
        ctx.lineWidth = 2;
        ctx.beginPath();
        ctx.moveTo(trail[i].x, trail[i].y);
        ctx.lineTo(trail[i+1].x, trail[i+1].y);
        ctx.stroke();
      }
    }

    // Draw rods
    ctx.strokeStyle = '#fff';
    ctx.lineWidth = 3;
    ctx.beginPath();
    ctx.moveTo(pivotX, pivotY);
    ctx.lineTo(x1, y1);
    ctx.stroke();
    ctx.beginPath();
    ctx.moveTo(x1, y1);
    ctx.lineTo(x2, y2);
    ctx.stroke();

    // Draw bobs
    ctx.fillStyle = '#ff6b6b';
    ctx.beginPath();
    ctx.arc(x1, y1, 10, 0, Math.PI*2);
    ctx.fill();
    ctx.fillStyle = '#4ecdc4';
    ctx.beginPath();
    ctx.arc(x2, y2, 10, 0, Math.PI*2);
    ctx.fill();

    // Add current lower bob position to trail
    trail.push({x: x2, y: y2});
    if (trail.length > maxTrailLength) {
      trail.shift();
    }
  }

  // Main loop
  let lastTime = 0;
  const dt = 0.02; // 50 Hz
  function animate(time) {
    if (!lastTime) lastTime = time;
    const elapsed = time - lastTime;
    if (!paused) {
      // Fixed time step integration
      while (elapsed >= dt) {
        state = rk4Step(state, dt);
        lastTime += dt;
      }
    }
    draw();
    requestAnimationFrame(animate);
  }

  // Event listeners
  pauseResumeBtn.addEventListener('click', () => {
    paused = !paused;
    pauseResumeBtn.textContent = paused ? 'Resume' : 'Pause';
    pauseResumeBtn.classList.toggle('paused', paused);
  });

  resetBtn.addEventListener('click', resetSimulation);

  // Start
  resetSimulation();
  requestAnimationFrame(animate);
</script>

Now we need to replace.

Let's do the edit.