When a harmonic oscillator's frequency ω(t) changes slowly, the adiabatic invariant J = E/ω is conserved. Rapid changes destroy invariance. Watch J as ω sweeps and track breakdown.
Phase Space (q, p)
Adiabatic Invariant J = E/ω
3.0×
0.050
1.5
—
J = E/ω now
—
J₀ initial
—
ΔJ/J₀ (%)
—
ω(t) current
Adiabatic Invariants (Classical Mechanics)
Action: J = ∮ p dq = E/ω (harmonic oscillator)
Adiabatic condition: |ω̇/ω²| ≪ 1 (slow change)
ω varies: amplitude scales as A ~ ω^(−1/2)
Quantum analog: n = J/ℏ = adiabatic quantum number (conserved)
Breakdown: resonance or rapid ramp → J no longer conserved