An adiabatic invariant J = ∮ p dq is a quantity conserved when a system parameter changes slowly compared to the oscillation period (adiabatic condition: ω̇ ≪ ω²). For a harmonic oscillator with slowly changing frequency, J = E/ω = const, so energy scales with frequency. This principle underlies the quantum adiabatic theorem (quantum numbers are preserved under slow parameter evolution) and plasma physics (the magnetic moment μ = mv²_⊥/2B is an adiabatic invariant, confining charged particles). The slowness of the sweep rate is critical — fast changes cause non-adiabatic transitions that violate J conservation.