Double Pendulum Chaos - Interactive Simulation

Interactive simulation of double pendulum chaotic motion with real-time visualization of chaotic behavior, phase space, and trajectory patterns

θ₁: 0.00°
θ₂: 0.00°
ω₁: 0.00 rad/s
ω₂: 0.00 rad/s
Time: 0.00 s

Phase Space (θ₁ vs ω₁)

Second Bob Trajectory

Energy Conservation

Kinetic: 0.00 J
Potential: 0.00 J
Total: 0.00 J

Parameters

Lagrangian Equations of Motion

Equation 1: (m₁+m₂)L₁θ₁'' + m₂L₂θ₂''cos(θ₁-θ₂) + m₂L₂θ₂'²sin(θ₁-θ₂) + (m₁+m₂)gsinθ₁ = 0
Equation 2: L₂θ₂'' + L₁θ₁''cos(θ₁-θ₂) - L₁θ₁'²sin(θ₁-θ₂) + gsinθ₂ = 0

What is a Double Pendulum?

A double pendulum consists of two pendulums attached end to end. Unlike a simple pendulum, the double pendulum exhibits chaotic behavior - small changes in initial conditions lead to dramatically different outcomes. This is a classic example of deterministic chaos in physics.

Chaos Theory

The double pendulum is a chaotic system, meaning it is highly sensitive to initial conditions. Even a tiny difference (0.001°) in the starting angle can result in completely different motion patterns after a few seconds. This is often called the "butterfly effect."

Phase Space

The phase space plot shows the relationship between position (θ) and velocity (ω). In chaotic systems, the phase space trajectory never repeats, forming complex patterns that demonstrate the system's unpredictable nature.

Energy Conservation

In the absence of friction, the total mechanical energy of the double pendulum is conserved. However, the energy continuously transforms between kinetic and potential forms in complex ways, contributing to the chaotic motion.