Double Pendulum - Chaos Theory

Time t 0.00 s

θ₁ 0.00°

θ₂ 0.00°

ω₁ 0.00 rad/s

ω₂ 0.00 rad/s

Kinetic T 0.00 J

Potential V 0.00 J

Total E 0.00 J

Divergence Time --

Launch 3 pendulums with tiny differences and watch how initial conditions amplify

Show Trail Trail Length 500

Length l₁ (m) 1.0 Length l₂ (m) 1.0 Mass m₁ (kg) 1.0 Mass m₂ (kg) 1.0 Gravity g (m/s²) 9.8 Damping Coefficient 0.00

θ₁ (rad) 1.57 θ₂ (rad) 1.57

Observe the system's trajectory in phase space

Phase Space Variable

The double pendulum is a classic application of Lagrangian mechanics. The system's Lagrangian is defined as:

L = T - V

where T is kinetic energy and V is potential energy

Coupled differential equations derived from Euler-Lagrange equations:

θ̈₁ = [m₂l₁ω₁²sinΔθ cosΔθ + m₂g sinθ₂ cosΔθ + m₂l₂ω₂²sinΔθ - (m₁+m₂)g sinθ₁] / [l₁(m₁+m₂) - m₂l₁cos²Δθ]

θ̈₂ = [-m₂l₂ω₂²sinΔθ cosΔθ + (m₁+m₂)(g sinθ₁ cosΔθ - l₁ω₁²sinΔθ - g sinθ₂)] / [l₂(m₁+m₂) - m₂l₂cos²Δθ]

where Δθ = θ₁ - θ₂

Nonlinear coupling: Two pendulums strongly coupled through trigonometric functions
Sensitive dependence: Tiny differences in initial conditions amplify exponentially
Energy conservation: System never repeats without damping