Chains of Kinematic Points

TL;DR

This paper proposes a new formulation for analyzing the stability of an infinite chain of cars by using the Banach space ll^, allowing for bounded but not necessarily vanishing perturbations, addressing limitations of the traditional Hilbert space approach.

Contribution

It introduces an alternative state space framework for stability analysis of car chains, enabling bounded perturbations without the vanishing condition.

Findings

Traditional ll^2 space imposes vanishing perturbations.

Using ll^ space allows bounded perturbations.

Addresses anomalous behaviors in classical formulations.

Abstract

In formulating the stability problem for an infinite chain of cars, state space is traditionally taken to be the Hilbert space $\ell^2$, wherein the displacements of cars from their equilibria, or the velocities from their equilibria, are taken to be square summable. But this obliges the displacements or velocity perturbations of cars that are far down the chain to be vanishingly small and leads to anomalous behaviour. In this paper an alternative formulation is proposed wherein state space is the Banach space $\ell^\infty$, allowing the displacements or velocity perturbations of cars from their equilibria to be merely bounded.