Adiabatic invariance derived from the Rund–Trautman identity and Noether’s theorem

Abstract

The generators of the infinitesimal transformations that lead to adiabatic invariance are derived from the Rund–Trautman identity by solving the Killing equations for a fairly generic class of models. Noether’s theorem then yields the conserved quantity, which for periodic motion with period T is 〈H〉T, where 〈H〉 is the time average of the Hamiltonian over one cycle. Further it is shown that if 〈H〉T is adiabatically invariant then so is the action ∮pdq, as the two differ by an invariant constant. Our approach (1) requires essentially no new concepts beyond those of a junior-level mechanics course, (2) shows how adiabatic invariance fits into the larger picture of the general connection between invariances and conservation laws, and (3) not only confirms and generalizes the results of Boltzmann, Clausius, and Ehrenfest for what is adiabatically invariant, but also predicts the rescaling transformations that lead to this type of invariance.