Journal Article

Symplectic Reduction and the Problem of Time in Nonrelativistic Mechanics

Karim P. Y. Thébault

in The British Journal for the Philosophy of Science

Published on behalf of British Society for the Philosophy of Science

Volume 63, issue 4, pages 789-824
Published in print December 2012 | ISSN: 0007-0882
Published online February 2012 | e-ISSN: 1464-3537 | DOI: https://dx.doi.org/10.1093/bjps/axr045
Symplectic Reduction and the Problem of Time in Nonrelativistic Mechanics

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The deep connection between the interpretation of theories invariant under local symmetry transformations (i.e. gauge theories) and the philosophy of space and time can be illustrated nonrelativistically via the investigation of reparameterization-invariant reformulations of Newtonian mechanics, such as Jacobi's theory. Like general relativity, the canonical formulation of such theories feature Hamiltonian constraints; and like general relativity, the interpretation of these constraints along conventional Dirac lines is highly problematic in that it leads to a nonrelativistic variant of the infamous problem of time. I argue that, nonrelativistically at least, the source of the problem can be found precsely within the symplectic reduction that goes along with strict adherence to the Dirac view. Avoiding reduction, two viable alternative strategies for dealing with Hamiltonian constraints are available. Each is found to lead us to a novel and interesting re-conception of time and change within nonrelativistic mechanics. Both these strategies and the failure of reduction have important implications for the debate concerning the relational or absolute status of time within physical theory.

1Introduction

2Mechanics with a Fixed Parameterization

  2.1Lagrangian mechanics

  2.2Hamiltonian mechanics

  2.3Symplectic mechanics

  2.4Presymplectic geometry and symplectic reduction

3Reductionism, Haecceitism, and Gauge Symmetry

4Reparameterization-Invariant Mechanics

  4.1Extended Lagrangian mechanics

  4.2Extended Hamiltonian mechanics

  4.3Jacobi's principle and timeless theory

  4.4Degeneracy, indeterminacy, and triviality

5Representing Change and Observables in Timeless Mechanics

  5.1The emergent time strategy

  5.2The correlation strategy

6Interpretational Implications

  6.1The relationalist versus substantivalist dispute with regard to time

  6.2An ontology of timeless change?

Journal Article.  14460 words.  Illustrated.

Subjects: Philosophy of Science ; Science and Mathematics

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