On symmetry and conserved quantities in classical mechanics


Authors
Jeremy Butterfield
Cambridge University
Abstract
This paper expounds the relations between continuous symmetries and conserved quantities, i.e. Noether's ``first theorem'', in both the Lagrangian and Hamiltonian frameworks for classical mechanics. This illustrates one of mechanics' grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem. I emphasise that, for both frameworks, the theorem is underpinned by the idea of cyclic coordinates; and that the Hamiltonian theorem is more powerful. The Lagrangian theorem's main ``ingredient'', apart from cyclic coordinates, is the rectification of vector fields afforded by the local existence and uniqueness of solutions to ordinary differential equations. For the Hamiltonian theorem, the main extra ingredients are the asymmetry of the Poisson bracket, and the fact that a vector field generates canonical transformations iff it is Hamiltonian.
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Laws and Meta-Laws of Nature: Conservation Laws and Symmetries.Marc Lange - 2007 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 38 (3):457-481.
Comparing Dualities and Gauge Symmetries.Sebastian De Haro, Nicholas Teh & Jeremy N. Butterfield - 2017 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 59:68-80.
On Classical Cloning and No-Cloning.Nicholas J. Teh - 2012 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 43 (1):47-63.

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