Lagrange's discrete model of the wave equation in dimension greater than one

TL;DR

This paper generalizes Lagrange's discrete string model of the wave equation from one dimension to higher dimensions, demonstrating that solutions can be approximated by mechanical models of coupled oscillators.

Contribution

It extends Lagrange's classical result to multiple spatial dimensions, providing a new discrete mechanical framework for the wave equation in higher dimensions.

Findings

Generalization of Lagrange's theorem to higher dimensions

Discrete mechanical models approximate wave solutions in multiple dimensions

Provides a foundation for numerical simulation of wave phenomena

Abstract

A celebrated theorem of Lagrange states that a solution of the wave equation with one-dimensional space variable is the uniform limit, as N tends to infinity, of a second order ODE obtained from a mechanical model discretizing a string as N identical harmonic oscillators. Answering to a question posed by G. Gallavotti we generalize this result to the case of any space dimension.