Two helpful developments towards better understanding of adiabatic invariance in classical mechanics

Abstract

The topic of difficult understanding of adiabatic invariance in classical mechanics is dealt with in a more understandable way. Using the one-dimensional harmonic oscillator as an example, the goals of this paper are twofold. First, given a first-order parameter variation, the second-order magnitude of the correction to the adiabatic invariant is established in simple terms. Second, the identification of the action variable with the invariant quantity for slow variations of different parameters of the Hamiltonian is confirmed, by invoking the correct equation of motion in the derivation.

Parametric invariance

“…The parametric invariance is included as a subject of classical mechanics [16][17][18][19][20][21][22], usually connected with the technique of action-angle variables. It is treated in quantum mechanics [23][24][25][26], kinetic theory and statistical mechanics [27][28][29], dynamics of charged particles [30][31][32][33], and has been applied to specific problems by several authors [34][35][36][37][38][39], particularly in modern computer calculations to determine entropy and free energy by the method of adiabatic switching [40,41] II. EHRENFEST PRINCIPLE Ehrenfest enunciated his hypothesis in 1916 in the following terms [12][13][14]: If a system be affected in a reversible adiabatic way, allowed motions are transformed into allowed motions.…”