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Preprints, Working Papers, ... Year : 2020

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

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.
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Dates and versions

hal-03070734 , version 1 (15-12-2020)
hal-03070734 , version 2 (17-12-2020)

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Thierry Gourieux, Raphaël Leone. On the interplay between Noether's theorem and the theory of adiabatic invariants. 2020. ⟨hal-03070734v2⟩
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