DOUBLE PENDULUM SIMULATION

A chaotic system of two connected pendulums demonstrating sensitive dependence on initial conditions. Small changes in starting positions lead to dramatically different trajectories.

CONTROL PANEL
10.0
10.0
2.0
2.0
45°
45°
9.81
SIMULATION VIEW
PHYSICS EQUATIONS

The double pendulum is described by coupled differential equations:

θ₁'' = (m₂g sin θ₂ cos(θ₁-θ₂) - m₂ sin(θ₁-θ₂)(l₁θ₁'² cos(θ₁-θ₂) + l₂θ₂'²) - (m₁+m₂)g sin θ₁) / (l₁(m₁ + m₂ sin²(θ₁-θ₂)))

θ₂'' = (m₂ sin(θ₁-θ₂)(l₁θ₁'² + g cos θ₁) + (m₁+m₂)(l₁θ₁'² sin(θ₁-θ₂) - g sin θ₂)) / (l₂(m₁ + m₂ sin²(θ₁-θ₂)))

Where θ₁, θ₂ are the angles, l₁, l₂ are the lengths, and m₁, m₂ are the masses.

REAL-TIME DATA
Angle 1: 0.00°
Angle 2: 0.00°
Angular Velocity 1: 0.00 rad/s
Angular Velocity 2: 0.00 rad/s
Total Energy: 0.00 J
Simulation Time: 0.00s
CHAOS THEORY

The double pendulum is a classic example of a chaotic system. It exhibits:

  • Sensitive Dependence on Initial Conditions: Tiny changes in starting positions lead to completely different trajectories
  • Deterministic Chaos: The system is completely predictable in theory, but practically unpredictable due to numerical precision limits
  • Strange Attractors: The system tends to explore certain regions of phase space more than others
  • Energy Conservation: Total mechanical energy remains constant (ignoring friction)

This demonstrates why long-term weather prediction is so difficult - small measurement errors compound exponentially over time.

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