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Chains of Kinematic Points

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.