The user wants me to create a single-page web app that simulates a chaotic double pendulum. This is a classic physics simulation that demonstrates chaotic behavior.

I need to:
1. Create an index.html file with CSS and JavaScript inline
2. Use the correct equations of motion for a double pendulum
3. Draw the pendulum arms on a canvas
4. Create a colorful fading trail for the tip
5. No external resources, CDNs, or network requests

Let me recall the equations of motion for a double pendulum:

A double pendulum consists of two masses m1 and m2 connected by rigid rods of lengths L1 and L2. The angles θ1 and θ2 are measured from the vertical.

The equations of motion (using Lagrangian mechanics) are quite complex. Here they are:

For θ1:
- θ1'' = (m2 * L1 * θ1'² * sin(θ1-θ2) * cos(θ1-θ2) + m2 * g * sin(θ2) * cos(θ1-θ2) + m2 * L2 * θ2'² * sin(θ1-θ2) - (m1+m2) * g * sin(θ1)) / ((m1+m2) * L1 - m2 * L1 * cos²(θ1-θ2))

For θ2:
- θ2'' = (-m2 * L2 * θ2'² * sin(θ1-θ2) * cos(θ1-θ2) + (m1+m2) * g * sin(θ1) * cos(θ1-θ2) - (m1+m2) * L1 * θ1'² * sin(θ1-θ2) - (m1+m2) * g * sin(θ2)) / ((m1+m2) * L2 - m2 * L2 * cos²(θ1-θ2))

These can be simplified. Let me use a more common form:

Let's define:
- num1 = -g(2m1+m2)sin(θ1) - m2*g*sin(θ1-2θ2) - 2sin(θ1-θ2)m2(θ2'²L2+θ1'²L1cos(θ1-θ2))
- den1 = L1(2m1+m2-m2cos(2θ1-2θ2))

θ1'' = num1/den1

For θ2:
- num2 = 2sin(θ1-θ2)(θ1'²L1(m1+m2) + g(m1+m2)cos(θ1) + θ2'²L2m2cos(θ1-θ2))
- den2 = L2(2m1+m2-m2cos(2θ1-2θ2))

θ2'' = num2/den2

I'll implement this with:
- A canvas element
- Physics simulation using these equations
- Trail system with fading colors
- Animation loop using requestAnimationFrame

Let me write this code.