Lecture 24: Aging spring and adiabatic invariants

We revisit the problem of a slowly-aging spring, in which the spring stiffness declines exponentially on a long time scale of O(1/epsilon) while a mass attached to the spring oscillates on a fast time scale of O(1). We previously solved the problem with the WKB method (Lecture 18); now we solve it with two-timing. New insights come from uncovering an "adiabatic invariant" for the system, a quantity that remains nearly constant for times up to and including the long time scale. In physical terms, the adiabatic invariant turns out not to be the system's total energy, but rather, its "action" defined as action = energy/oscillation frequency. We also look at animations of a mass vibrating on an aging spring (courtesy of Arnaldo Gonzalez-Rodriguez, a former student in this course) and briefly discuss how the insights about adiabatic invariants informed early work by Planck and Einstein on the "old quantum theory" (pre-Heisenberg and Schrödinger).