Fast forward to the classical adiabatic invariant

We show how the classical action, an adiabatic invariant, can be preserved under non-adiabatic conditions. Specifically, for a time-dependent Hamiltonian $H=p^2\mathrm{/}2m+U(q,t)$ in one degree of freedom, and for an arbitrary choice of action $I_0$, we construct a “fast-forward” potential energy function $V_{\rm FF}(q,t)$that,whenaddedto$H$,guidesalltrajectorieswithinitialaction$I_0$ to end with the same value of action. We use this result to construct a local dynamical invariant $J(q,p,t)$ whose value remains constant along these trajectories. We illustrate our results with numerical simulations. Finally, we sketch how our classical results may be used to design approximate quantum shortcuts to adiabaticity.