 Solution

An adiabatic invariant remains constant to leading order when a system parameter changes on a timescale much longer than one period. For a periodic one-degree-of-freedom system, the adiabatic invariance of the action supplies such an invariant.

The momentum magnitude between collisions is $p_0=\sqrt{2mE}$. A complete orbit crosses the gap once in each direction, so the unnormalized action used in the question isConsequentlybecause the speed is $\sqrt{2E/m}$ and the round-trip distance is $2L$.

When $L(t)$ varies adiabatically, conservation of the adiabatic particle between moving parallel walls action givesA moving wall does work at each collision: an approaching wall raises the particle's energy and a receding wall lowers it. Elasticity holds in the instantaneous rest frame of the wall.