Abstract

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?

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