Prompt training double pendulum implementation

Now I have full context. Here's the implementation plan:

Implementation Plan: "Prompt Training: Double Pendulum"

Libraries Check ✅

• No new libs needed. The double pendulum is naturally 2D, so we'll use the native Canvas API (no Three.js). Math rendering via existing Math component (rehype-katex/remark-math).

Files to Create

1. src/components/DoublePendulumVisualization.astro — Interactive component

   • 2D Canvas rendering the pendulum arms, bobs, and trailing path
   • Controls panel with sliders for: L₁, L₂, m₁, m₂, θ₁, θ₂
   • RK4 integrator (Euler is too unstable for this system)
   • Trail of the second bob to reveal chaotic patterns
   • Play/Pause & Reset buttons
   • Dark theme styling matching Lorenz (background #0a0a0f, cyan accents)

2. src/content/blog/double-pendulum.mdx — Blog post

   • Frontmatter: title, description, pubDate, aiGenerated: true
   • Intro: what a double pendulum is, why it's chaotic
   • Equations: Lagrangian-derived equations of motion (using <Math>)
   • The interactive visualization with controls
   • Python code example
   • Discussion of sensitivity to initial conditions

Physics

The double pendulum uses Lagrangian mechanics with generalized coordinates (θ₁, θ₂). The equations of motion involve coupled second-order ODEs — we'll implement them as a 4-variable first-order system [θ₁, ω₁, θ₂, ω₂] integrated with RK4 at a small timestep.

Want me to proceed with the implementation?

Let me check the Math component and global styles to fully understand the patterns before writing code.

Good — KaTeX CSS is loaded globally. Now I have full context. Here's my todo list, then I'll implement step by step:

1. ✅ Understand existing patterns (Lorenz component, blog post format, libs)
2. Create src/components/DoublePendulumVisualization.astro — Canvas + controls + RK4 physics
3. Create src/content/blog/double-pendulum.mdx — Blog post content

Let me start with step 2 — the visualization component:

Now step 3 — the blog post:

Now let me build and verify:

Build succeeds. Let me deploy:

Everything built and deployed successfully. The new blog post is live at https://robertsoare.xyz/blog/double-pendulum/ with:

• 2D Canvas double pendulum simulation with RK4 integration
• 6 interactive sliders: L₁, L₂ (rod lengths), m₁, m₂ (masses), θ₁, θ₂ (starting angles)
• Play/Pause & Reset buttons — adjust sliders then press Reset to restart with new parameters
• Cyan trail on the lower bob showing the chaotic path
• Glowing bobs with size proportional to mass (red = upper, cyan = lower)
• Full blog content with Lagrangian equations, Python RK4 code, and chaos discussion