Action variable conservation under slow parameter change in a pendulum
The adiabatic invariant J = ∮ p dq (loop integral in phase space) is conserved when a system parameter changes slowly compared to the oscillation period. For a pendulum of length L: J = ∮ p dθ ∝ E/ω ∝ E·√L. As L changes slowly, E changes but J stays constant. This underlies quantum adiabatic theorem (ℏ·n = J), magnetic moment conservation, and the basis of action-angle variables in Hamiltonian mechanics.