Noether’s theorem, the Rund–Trautman function, and adiabatic invariance

Abstract

This article focuses on the recognition of an important quantity that will be called the Rund–Trautman function. It already plays a central role in Noether’s theorem since its vanishing characterizes a symmetry and leads to a conservation law. The main aim of the paper will be to show how, in the realm of classical mechanics, an ‘almost’ vanishing Rund–Trautman function accompanying an ‘almost’ symmetry leads to an ‘almost’ constant of motion, especially within the adiabatic hypothesis for which the ‘almostness’ in question is in some sense measured by the slowness of time-dependent parameters. To this end, the Rund–Trautman function is first introduced and analyzed in detail, then it is implemented for the general one-dimensional problem. Finally, its relevance in the adiabatic context is examined through the example of the harmonic oscillator with a slowly varying frequency. Notably, for some frequency profiles, explicit expansions of adiabatic invariants are derived through it and an illustrative numerical test is realized.