Anharmonic oscillator: A playground to get insight into renormalization

TL;DR

This paper explores how interactions affect the frequency of anharmonic oscillators, using quantum field theory-inspired methods, and examines frequency shifts and beat phenomena in coupled nonlinear oscillators.

Contribution

It reviews existing techniques and introduces new methods for analyzing frequency renormalization in anharmonic oscillators, including multi-oscillator systems.

Findings

Frequency deviations due to interactions are analyzed.

New computational techniques for frequency shifts are proposed.

Beat phenomena in coupled oscillators are observed and explained.

Abstract

In the presence of interactions the frequency of a simple harmonic oscillator deviates from the noninteracting one. Various methods can be used to compute the changes to the frequency perturbatively. Some of them resemble the methods used in quantum field theory where the bare values of the parameters of the theory run when an interaction is added. In this paper, we review some of these techniques and introduce some new ones in line with quantum field theory methods. Moreover, we investigate the case of more than one oscillator to see how the frequencies of small oscillations change when non-linear terms are added to a linear system and observe an interesting beat phenomenon in degenerate coupled oscillatory systems.