One dimensional fields and adiabatic invariance

The adiabatic invariance properties of the single particle motion are studied for a class of one-dimensional fields. If the fields are sufficiently smooth the theory of the(non-linear) one-dimensional oscillator applies, which immediately leads to the result that the action integral J = $\phi$pdq is an adiabatic invariant to all orders in $\epsilon$, the reciprocal time scale of the field. Some situations of physical interest, however, require a special treatment because the fields show some lack of smoothness. Two examples are discussed: Fields having jump discontinuities and small scale fields superimposed on a constant magnetic field. In both cases adiabatic invariance of J is established. The results are discussed from several physical points of view as plane collision-free shocks, single particle acceleration, euiding center stability, and adiabatic compression