Two Coupled Oscillators — Central Force

Two coupled oscillators

U₁ = ½kr₁² · U₂ = ½kr₂² · U₁₂ = ½αkr²

Parameters

k (stiffness) 1.00

α (coupling) 1.00

mass m 1.00

CM initial conditions

X₀ 1.00

Ẋ₀ 0.00

Y₀ 0.00

Ẏ₀ 1.00

Relative coordinate initial conditions

x₀ 0.80

ẋ₀ 0.00

y₀ 0.00

ẏ₀ 0.50

Playback

Speed 1.0×

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Display

Trail — particle 1 Trail — particle 2 CM trajectory Relative coord trajectory Grid & origin Spring between particles

Live stats

ω_cm

—

ω_rel

—

|R|

—

|r|

—

E_cm

—

E_rel

—

Analytical solution

// CM decoupled oscillator ω²_cm = k/m X(t) = X₀ cos(ω_cm t) + (Ẋ₀/ω_cm) sin(ω_cm t) Y(t) = Y₀ cos(ω_cm t) + (Ẏ₀/ω_cm) sin(ω_cm t) // Relative coord oscillator ω²_rel = (1+2α)k/m x(t) = x₀ cos(ω_rel t) + (ẋ₀/ω_rel) sin(ω_rel t) y(t) = y₀ cos(ω_rel t) + (ẏ₀/ω_rel) sin(ω_rel t) // Recover particles r₁ = R + r/2 r₂ = R - r/2

Legend

Particle 1 (m₁)

Particle 2 (m₂)

Centre of mass R

Relative coord midpoint