Double Pendulum Laboratory Chaos Instrument No. 7

RK4 · Δt 1/240 s · Lagrangian

t = 0.00 s

E = 0.000 J

Drag either bob to set initial angles · release to let go

Parameters

mass m₁1.00 kg

mass m₂1.00 kg

length ℓ₁1.00 m

length ℓ₂1.00 m

gravity g9.81 m/s²

damping γ0.000

trail length600 pts

Readout

θ₁2.094

θ₂2.094

ω₁0.000

ω₂0.000

Steps0

Divergence—

Notebook

Equations of motion, from the Lagrangian L = T − V:

θ₁ = −g(2m₁+m₂) sin θ₁ − m₂ g sin(θ₁−2θ₂) − 2 sin Δ · m₂(θ₂²ℓ₂ + θ₁²ℓ₁ cos Δ)ℓ₁(2m₁ + m₂ − m₂ cos 2Δ)

θ₂ = 2 sin Δ · (θ₁²ℓ₁(m₁+m₂) + g(m₁+m₂) cos θ₁ + θ₂²ℓ₂m₂ cos Δ)ℓ₂(2m₁ + m₂ − m₂ cos 2Δ)

where Δ = θ₁ − θ₂, damping −γθ_i

Integrated with fourth-order Runge–Kutta:

y_n+1 = y_n + h6(k₁ + 2k₂ + 2k₃ + k₄)

Two systems separated by 0.001 rad in θ₁ diverge exponentially — the signature of deterministic chaos. Watch Δ_bob grow from millimetres to metres in seconds.