Adiabatic invariant

Summary

An adiabatic invariant is a quantity of a mechanical system that stays nearly constant under a very slow change of a parameter. Here slow means slow compared with the motion itself. It is not exactly conserved, but its total change over the long time of the slow variation tends to zero as the variation becomes slower. For a periodic motion with one degree of freedom, the action $\frac{1}{2\pi} \oint p \, dq$ is the basic adiabatic invariant; see action-angle coordinates. Adiabatic invariants explain, for example, the trapping of charged particles in magnetic mirrors and appear in the old quantum theory. See the definition.