J = ∮ p dq ≈ const when parameters change slowly
For a pendulum of slowly varying length L(t), the adiabatic invariant:
where ω = √(g/L). As L shrinks, ω↑ and amplitude grows such that E/ω stays fixed. Energy is not conserved — J is.
This is the classical analogue of quantum number conservation during slow Hamiltonian changes (Born-Fock theorem).