Symplectic reduction and the problem of time in nonrelativistic mechanics

Karim P Y Thébault*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

7 Citations (Scopus)


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.1 Introduction2 Mechanics with a Fixed Parameterization 2.1 Lagrangian mechanics 2.2 Hamiltonian mechanics 2.3 Symplectic mechanics 2.4 Presymplectic geometry and symplectic reduction3 Reductionism, Haecceitism, and Gauge Symmetry4 Reparameterization- Invariant Mechanics 4.1 Extended Lagrangian mechanics 4.2 Extended Hamiltonian mechanics 4.3 Jacobi's principle and timeless theory 4.4 Degeneracy, indeterminacy, and triviality5 Representing Change and Observables in Timeless Mechanics 5.1 The emergent time strategy 5.2 The correlation strategy6 Interpretational Implications 6.1 The relationalist versus substantivalist dispute with regard to time 6.2 An ontology of timeless change?

Original languageEnglish
Pages (from-to)789-824
Number of pages36
JournalBritish Journal for the Philosophy of Science
Issue number4
Publication statusPublished - Dec 2012


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