Investigation of Power-Law Damping/Dissipative Forces

TL;DR

This paper explores unique power-law dissipative forces in one-dimensional particle systems, revealing forces that cause the particle to stop in finite time or only after infinite travel, expanding understanding of dissipative dynamics.

Contribution

It introduces novel power-law dissipative forces with distinct stopping behaviors, contrasting with traditional linear damping, and discusses implications for complex systems.

Findings

Some forces cause particles to stop in finite time

Other forces lead particles to reach zero velocity only after infinite distance

Results suggest new types of dissipative behaviors in dynamical systems

Abstract

The properties of a one space-dimension, one particle dynamical system under the influence of a purely dissipative force are investigated. Assuming this force depends only on the velocity, it is demonstrated, in contrast to the case of linear damping, that there exist dissipative forces for which the particle \textquotedblleft stops" in a finite time. It is also shown, by an explicit example, that other dissipative forces exist such that they produce dynamics in which the particle achieves zero velocity only after an infinite distance has been traveled. Possible applications of these results to more complex situations are discussed.