an adiabatic invariant
An adiabatic invariant is a quantity that stays almost perfectly constant when you change a system's parameters slowly, even though it is not exactly conserved. Picture a pendulum whose string you shorten very gradually. Its energy changes and its frequency changes, but a particular combination -- the action -- barely moves. Slow change, it turns out, protects certain quantities in a way that sudden change does not.
For a system executing periodic motion, the adiabatic invariant is the action J = (1/2 pi) times the closed-loop integral of p dq over one cycle -- the phase-space area enclosed by the orbit, divided by 2 pi. The theorem is: if a parameter lambda of the Hamiltonian is varied slowly compared with the orbital period (so that its fractional change per period is tiny), then J is conserved to exponentially good accuracy, dJ/dt being of higher order in the slowness. Energy and amplitude are free to drift; the enclosed phase-space area is what nature holds fixed. 'Adiabatic' here means slow relative to the internal motion, not (as in thermodynamics) merely no-heat-flow, though the two senses are historically linked.
Adiabatic invariants are workhorses across physics. For a slowly shortened pendulum, J = E/omega constant means E grows in proportion to omega (Einstein's example at the 1911 Solvay conference). A charged particle spiraling in a slowly varying magnetic field keeps its magnetic moment mu nearly constant, which is what confines plasmas in magnetic mirrors and traps particles in Earth's radiation belts. Historically the adiabatic invariance of the action is exactly what made Bohr-Sommerfeld quantization (J = n h) consistent: quantum numbers cannot change under slow perturbations, so they are good labels.
A pendulum of length L slowly shortened has frequency omega = sqrt(g/L) and energy E. The action J = E/omega is adiabatically conserved, so E is proportional to omega, i.e. E proportional to L^(-1/2). Halve the length slowly and the energy rises by a factor 2^(1/4) -- energy fed in by the hand doing work against the string tension, bookkept exactly by the constancy of the action.
Slowly shortening a pendulum conserves the action J = E/omega, forcing its energy to rise as the frequency rises.
Adiabatic invariance requires the change to be slow compared with the system's own period, and it is approximate, not exact -- the action can change if the driving is fast, or across a separatrix where the period diverges (the pendulum passing from swinging to going over the top), where adiabatic invariance breaks down entirely.