Why is the motion chaotic?

The system is nonlinear and has more than one degree of freedom with coupling, so most initial conditions lead to long-term behavior that is extremely sensitive to small perturbations.

Double Pendulum

Two coupled rods or masses: a classic chaotic system. Tiny changes in initial angles produce diverging trajectories, illustrating sensitive dependence on initial conditions.

Who it's for: Advanced high school or undergraduate; nonlinear dynamics and chaos demos.

Key terms

double pendulum
chaos
coupled oscillators
phase space
nonlinearity

Live graphs

How it works

A classic chaotic system: two point masses on massless rods. Tiny changes in initial angles produce wildly different trajectories. RK4 integration with optional light damping keeps long runs stable.

Key equations

Coupled nonlinear ODEs for θ₁(t), θ₂(t); see standard references for explicit θ¨₁, θ¨₂ (Wikipedia: double pendulum).

E = T + V with T from rod-end velocities and V = −m₁gL₁cos θ₁ − m₂g(L₁cos θ₁ + L₂cos θ₂) + const.

Frequently asked questions

Why is the motion chaotic?: The system is nonlinear and has more than one degree of freedom with coupling, so most initial conditions lead to long-term behavior that is extremely sensitive to small perturbations.