The Adiabatic Invariance of the Action Variables

Abstract

We shall first use an example to explain the concept of adiabatic invariance. Let us consider a “super ball” of mass m, which bounces back and forth between two walls (distance l) with velocity $\boldsymbol{v}_{0}$. Let gravitation be neglected, and the collisions with the walls be elastic. If F _m denotes the average force onto each wall, then we have

$$\displaystyle{ F_{m}T = -\int _{\mathrm{coll.\,time}}f\,dt\;. }$$

(9.1)

f is the force acting on the ball during one collision, and T is the time between collisions.