Adiabatic Invariant

When a pendulum's length changes slowly compared to its period (ε ≪ 1), the action J = ∮p dq ≈ E/ω is conserved. Energy changes but the ratio E/ω remains constant. This is the classical adiabatic theorem — a precursor to quantum adiabatic processes.

Initial length L₀: 1.00 m

Final length L₁: 0.40 m

Slowness ε: 0.02

Initial angle θ₀: 30°

L(t): — m

E(t): — J

ω(t): — rad/s

J=E/ω: —

J₀=E₀/ω₀: —

J error: —

Adiabatic: ε = (1/ω)(dω/dt) ≪ 1
Action: J = ∮p·dq = E/ω
dJ/dt = O(ε) → J conserved
E ~ √(g/L) → E·√L = const

Adiabatic Invariant — Pendulum