Double pendulum · minimax-m2

The user wants me to create a double pendulum simulation with:
1. Correct equations of motion for a chaotic double pendulum
2. Drawing the arms (two connected rods)
3. A colorful fading trail of the tip

This is a physics simulation that needs to use the Lagrangian mechanics equations for a double pendulum. Let me create a single HTML file with:
- Canvas for rendering
- JavaScript for the physics simulation
- CSS for styling

The double pendulum equations of motion are:
For a double pendulum with masses m1, m2, lengths l1, l2, and angles θ1, θ2:

The accelerations can be derived from the Lagrangian:
- θ1'' = [-g(2m1+m2)sin(θ1) - m2*g*sin(θ1-2θ2) - 2sin(θ1-θ2)*m2*(θ2'^2*l2 + θ1'^2*l1*cos(θ1-θ2))] / [l1*(2m1+m2-m2*cos(2θ1-2θ2))]

- θ2'' = [2sin(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))]

Let me create this simulation.

I've created a complete double pendulum simulation with:

1. **Correct equations of motion** - Using the Lagrangian mechanics derivation for a double pendulum with Runge-Kutta 4th order integration for numerical stability

2. **Visual elements**:
   - Two arms drawn as gradient lines (blue to purple to pink)
   - Two bobs with glowing radial gradients
   - A colorful fading trail that cycles through hues
   - A pivot point with glow effect

3. **Interactive features**:
   - Click and drag either bob to reposition the pendulum
   - Reset button to start with new random initial conditions
   - Pause/Resume button
   - Trail clears when you drag

4. **Styling**:
   - Dark background
   - Neon-style colors (pink, purple, blue)
   - Glow effects using shadows and gradients
   - Responsive feel

The physics uses the correct mathematical formulas:
- Angular accelerations computed from the Lagrangian
- Proper handling of the coupled differential equations
- Substepping for numerical stability
- Very slight damping for realism

The file is fully self-contained with no external dependencies.