On the interplay between Noether's theorem and the theory of adiabatic invariants

TL;DR

This paper explores how the Rund-Trautman function, related to Noether's theorem, can be used to understand adiabatic invariants in classical mechanics, especially in systems like the harmonic oscillator with slowly varying parameters.

Contribution

It introduces and analyzes the Rund-Trautman function in the context of adiabatic invariants, providing explicit expansions and numerical tests for specific systems.

Findings

The Rund-Trautman function characterizes near-symmetries and near-conservation laws.

Explicit adiabatic invariants are derived for certain frequency profiles.

Numerical tests confirm the theoretical predictions.

Abstract

This article focuses on 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 is 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 within the adiabatic assumption, that is, to an adiabatic invariant. To this end, the Rund-Trautman function is first introduced and analysed 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.