What is a Newtonian system? The failure of energy conservation and determinism in supertasks
type: Item
instance of: scholarly article
title: What is a Newtonian system? The failure of energy conservation and determinism in supertasks
zbMATH Open document ID: 1015.70005
DOI: 10.1023/A:1005257429769
published in: Synthese
publication date: 2002-10-31T00:00:00Z
review text: The ``supertask'' (ST) is defined by infinitely many unit point masses \(P_1,P_2,\cdots\) arrayed at rest along the \(x\)-axis of space at positions \(x=\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \cdots\). At time \(t=0\) a unit point mass \(P_o\), moving at unit velocity, passes \(x=0\) and approaches the remaining masses. Infinitely many perfectly elastic collisions ensue with the unit velocity passing in succession from \(P_o\) to \(P_1\), from \(P_1\) to \(P_2\) etc. However, at \(t=1\), all collisions are completed, and all masses are at rest. The mathematical description of the process can be found in chapter 3 (``Choosing a mathematical model for the supertask''), which reveals the results of an assumed dynamical process without explaining the process itself explicitly: In eq. (2.1) (Newton's second law), a distiction between \(F_i=F_i^i+F_i^r\) (impressed forces and reaction forces) for the beginning of a contact is missing. With an interpretation of eq. (2.2) as d'Alembert's (original) principle (i.e. the reaction forces \(F_i^r\) equilibrate), the underlying dynamical process might be looked at as implicitly defined with \(F_i^i=0\) for all \(i\) and \(\sum F_i^r = 0\). The system under consideration moves then due to an initial condition (\(P_o\)) and undergoes constraints leading to ``instantaneous velocity changes''. These would, however, already ``threaten the Newtonian credentials'' because of its discontinuity. ``The most natural solution is to assume that each collison is mediated by very strong short range forces that eliminate the discontinuities in the time dependence of velocities and provide them sufficient differentiability for the acceleration to be always defined''. In that case, eq. (2.2) may be interpreted in the authors' sense (``Newton's third law'') as a summation over the impressed forces \(\sum F_i^i = 0\) which indicates that the impressed forces are inner forces and no additional constraints occur. D'Alembert's principle then turns to t…
reviewed by: Hartmut Bremer
Mathematics Subject Classification ID: 70A05
zbMATH DE Number: 1563711
zbMATH Keywords: determinism
zbMATH Keywords: supertask
zbMATH Keywords: unit point masses
zbMATH Keywords: perfectly elastic collisions
zbMATH Keywords: Newton's second law
zbMATH Keywords: d'Alembert principle
zbMATH Keywords: reaction forces
zbMATH Keywords: discontinuity
zbMATH Keywords: Newton's third law
zbMATH Keywords: Lagrangian parameter
zbMATH Keywords: Newton's impact law
zbMATH Keywords: conservation of energy
author: Joseph S. Alper
author: Mark Bridger
author: John Earman
author: John D. Norton
MaRDI profile type: Publication
full work available at URL: a:1005257429769
OpenAlex ID: W2164975880
description: scientific article; zbMATH DE number 1563711