The action J = ∮p dq is conserved when parameters change slowly
Adiabatic Invariants: For a harmonic oscillator with slowly varying frequency ω(t), the action J = ∮p dq = E/ω is conserved to all orders in ε = (dω/dt)/ω² ≪ 1. As ω increases, the energy E = J·ω increases proportionally — like a pendulum whose length is slowly shortened. In the phase space (q,p), the orbit area = 2πJ stays constant even as the ellipse changes shape. This is the classical analog of quantum adiabatic theorem (energy eigenstate follows eigenstate as H changes slowly). Breakdown occurs when ε ~ 1 (fast sweep): non-adiabatic transitions cause J to change. The action variable is the zeroth moment of the Poincaré invariant.