Why is the motion chaotic?

The system is nonlinear and has two degrees of freedom with strong coupling, so most large-amplitude initial conditions lead to long-term behavior that is extremely sensitive to small perturbations — nearby trajectories diverge in phase space.

Why does total energy E drift even with zero damping?

RK4 is not a symplectic integrator. Truncation error slowly changes the numerical Hamiltonian, so ΔE and ΔE/E₀ can creep even when the continuous model conserves energy. Shorter steps reduce the drift; long chaotic runs accumulate more.

What does the angular damping slider do?

It applies a simple multiplicative decay ω ← ω e^(−γ Δt) after each RK4 substep. That is pedagogical dissipation, not a fix for numerical instability. With γ > 0, mechanical energy falls on purpose; with γ = 0 you are watching integrator drift alone.

How does small-angle motion differ from chaos?

For small angles the equations linearize into coupled harmonic oscillators and motion stays nearly periodic (try the Small-angle preset). Large angles keep the full nonlinear coupling, and trajectories typically become irregular and sensitive to initial conditions (Classic chaos preset).

Is tip separation the Lyapunov exponent?

No. The compare mode integrates a second copy with θ₁ offset by 0.1° and reports the Euclidean distance between the two lower bobs. That visualizes sensitive dependence; a Lyapunov exponent λ would require a carefully normalized logarithmic growth rate of infinitesimal perturbations over long time.

Double Pendulum

Mesmerizing chaotic motion with path tracing.

Live graphs

About this model

Two point masses on massless rods form a classic chaotic system. Angles θ₁, θ₂ are measured from the downward vertical; the coupled nonlinear ODEs for θ¨₁ and θ¨₂ (standard Wikipedia / textbook form with δ = θ₁ − θ₂) are integrated with RK4. Mechanical energy E = T + V uses rod-end kinetic energy and V = −m₁gL₁ cos θ₁ − m₂g(L₁ cos θ₁ + L₂ cos θ₂). Optional angular damping multiplies ω after each step. An optional second trajectory starts with θ₁ shifted by 0.1° so you can watch tip paths diverge — that separation is not a Lyapunov exponent.

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

Key terms

double pendulum
chaos
coupled oscillators
phase space
nonlinearity
RK4
energy drift

How it works

A classic chaotic system: two point masses on massless rods. Tiny changes in initial angles produce wildly different trajectories. RK4 integration; optional angular damping dissipates energy (it is not a numerical stabilizer).

Key equations

With δ = θ₁ − θ₂ and den = 2m₁ + m₂ − m₂ cos(2δ): θ¨₁ = [−g(2m₁+m₂)sin θ₁ − m₂ g sin(θ₁−2θ₂) − 2 sin δ · m₂(ω₂² L₂ + ω₁² L₁ cos δ)] / (L₁ den); θ¨₂ = [2 sin δ · (ω₁² L₁(m₁+m₂) + g(m₁+m₂)cos θ₁ + ω₂² L₂ m₂ cos δ)] / (L₂ den). Angles from downward vertical.

E = T + V: T from rod-end velocities; V = −m₁gL₁ cos θ₁ − m₂g(L₁ cos θ₁ + L₂ cos θ₂) (+ const). Optional damping multiplies ω by e^(−γ Δt) after each RK4 step.

Frequently asked questions

Why is the motion chaotic?: The system is nonlinear and has two degrees of freedom with strong coupling, so most large-amplitude initial conditions lead to long-term behavior that is extremely sensitive to small perturbations — nearby trajectories diverge in phase space.
Why does total energy E drift even with zero damping?: RK4 is not a symplectic integrator. Truncation error slowly changes the numerical Hamiltonian, so ΔE and ΔE/E₀ can creep even when the continuous model conserves energy. Shorter steps reduce the drift; long chaotic runs accumulate more.
What does the angular damping slider do?: It applies a simple multiplicative decay ω ← ω e^(−γ Δt) after each RK4 substep. That is pedagogical dissipation, not a fix for numerical instability. With γ > 0, mechanical energy falls on purpose; with γ = 0 you are watching integrator drift alone.
How does small-angle motion differ from chaos?: For small angles the equations linearize into coupled harmonic oscillators and motion stays nearly periodic (try the Small-angle preset). Large angles keep the full nonlinear coupling, and trajectories typically become irregular and sensitive to initial conditions (Classic chaos preset).
Is tip separation the Lyapunov exponent?: No. The compare mode integrates a second copy with θ₁ offset by 0.1° and reports the Euclidean distance between the two lower bobs. That visualizes sensitive dependence; a Lyapunov exponent λ would require a carefully normalized logarithmic growth rate of infinitesimal perturbations over long time.