The classical harmonic chain: solution via Laplace transforms and continued fractions

TL;DR

This paper presents an exact analytical method using Laplace transforms and continued fractions to solve the classical harmonic chain, providing clear insights into particle displacements and connecting to quantum Green function techniques.

Contribution

It introduces a novel approach to solving the harmonic chain using Laplace transforms and continued fractions, offering an analytical evaluation of displacements.

Findings

Displacements expressed as Bessel functions

Continued fraction representation of Laplace transforms

Analytical solutions applicable to simple harmonic chains

Abstract

The harmonic chain is a classical many-particle system which can be solved exactly for arbitrary number of particles (at least in simple cases, such as equal masses and spring constants). A nice feature of the harmonic chain is that the final result for the displacements of the individual particles can be easily understood -- therefore, this example fits well into a course of classical mechanics for undergraduates. Here we show how to calculate the displacements by solving equations of motion for the Laplace transforms $\mathcal{L}\left\{q_n\right\}(s)$ of the displacements $q_n(t)$. This leads to a continued fraction representation of the Laplace transforms which can be evaluated analytically. The inverse Laplace transform of $\mathcal{L}\left\{q_n\right\}(s)$ finally gives the displacements which generically have the form of Bessel functions. We also comment on the similarities between this approach and the Green function method for quantum many-particle systems.