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A quantity conserved to leading order when a parameter varies slowly compared with the natural motion period. The action variable of a regular periodic Hamiltonian orbit is a standard example, as explained by adiabatic invariance of the action.

Liouville integrability on a $2n$-dimensional symplectic manifold means that there are $n$ functionally independent first integrals, including $H$, whose pairwise Poisson brackets vanish. Near a compact connected regular common level set, the Arnold-Liouville theorem gives action-angle variables $(J_i,{\theta }_i)$ with $J_i=(2\pi {)}^{-1}\oint pdq$, ${\theta }_i$ periodic modulo $2\pi$, $H=H(J)$, ${\dot{J}}_i=0$ and ${\dot{\theta }}_i=\partial H/\partial J_i$.

For a frozen mass, the turning point is $q_{\ast }=(2E{)}^{1/(2k)}$. The action variable isAdiabatic invariance of the action holds to leading order when the mass varies slowly compared with an orbital period, away from degenerating orbits. Keeping $J$ fixed givesFor $k=1$ this agrees with $E=\omega J$ and $\omega \propto m^{-1/2}$.

The adiabatic invariance of the action applies when the frequency changes by only a small fraction during one oscillation and no separatrix is encountered. A convenient condition isThenConsequentlyIf $A$ is the displacement amplitude, $E=\frac{1}{2}k{A}^2$, and henceThus a slowly stiffened spring raises the energy as $k^{1/2}$ while reducing the amplitude as $k^{-1/4}$.

Put $\theta =k\log t$. Sincethe instantaneous oscillator energy isThe instantaneous frequency is $\omega (t)=1/(ϵt)$, soIts oscillatory part divided by its mean is $O(k^{-1})=O(ϵ)$. It remains bounded and periodic in $\log t$, so the variations do not accumulate. This is the expected behavior of an adiabatic invariant: slow parameter variation produces small bounded oscillations rather than secular drift.

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.