Distinguished Limits and Vibrogenic Force revealed by Newton's Equation with Oscillating Force

TL;DR

This paper analyzes Newton's equation with high-frequency oscillating force, revealing two distinguished asymptotic limits and introducing the universal vibrogenic force, providing a simple, general approach applicable to various oscillatory differential equations.

Contribution

It introduces a simple, general analysis of distinguished limits and vibrogenic force in oscillatory Newton's equations, offering a practical guide for various differential equations.

Findings

Identified two distinguished limits in high-frequency oscillations.

Discovered the universal vibrogenic force as a key component in averaged equations.

Provided a straightforward method applicable to ODEs and PDEs with oscillating coefficients.

Abstract

In this paper, we analyse the basic ideas of Vibrodynamics and the two-timing method. To make our analysis most instructive, we have chosen the Newton's equation with a general oscillating force. We deal with its asymptotic solutions in the high frequency limit. Our treatment is simple but general. The targets of our study are \emph{the distinguished limits} and \emph{the universal vibrogenic force}. The aim of \emph{the distinguished limit procedure} is to identify how the small parameter can appear in an equation. The proper appearance of a small parameter leads to \emph{valid successive approximations}, and, in particular, to closed systems of averaged equations. We show, that there are only two distinguished limits. This means that Newton's equation, with high-frequency forcing, has two types of interesting asymptotic solutions. The key item in the averaged equations for all distinguished limits is \emph{the unique vibrogenic force}. The current state-of-the-art in this area is: a large number of particular examples are well-known, effective and advanced general methods (like the Krylov-Bogolyubov approach) are well developed. However, the presented general and simple analysis of distinguished limits and the vibrogenic force, formulated as a compact practical guide, is novel. An advantage of our treatment is the possibility of its straightforward use for various ODEs and PDEs with oscillating coefficients.