dorsal/arxiv
View SchemaOn Symmetry and Conserved Quantities in Classical Mechanics
| Authors | Jeremy Butterfield |
|---|---|
| Categories | |
| ArXiv ID | physics/0507192 |
| URL | https://arxiv.org/abs/physics/0507192 |
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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"abstract": "This paper expounds the relations between continuous symmetries and conserved\nquantities, i.e. Noether\u0027s ``first theorem\u0027\u0027, in both the Lagrangian and\nHamiltonian frameworks for classical mechanics. This illustrates one of\nmechanics\u0027 grand themes: exploiting a symmetry so as to reduce the number of\nvariables needed to treat a problem.\n I emphasise that, for both frameworks, the theorem is underpinned by the idea\nof cyclic coordinates; and that the Hamiltonian theorem is more powerful. The\nLagrangian theorem\u0027s main ``ingredient\u0027\u0027, apart from cyclic coordinates, is the\nrectification of vector fields afforded by the local existence and uniqueness\nof solutions to ordinary differential equations. For the Hamiltonian theorem,\nthe main extra ingredients are the asymmetry of the Poisson bracket, and the\nfact that a vector field generates canonical transformations iff it is\nHamiltonian.",
"arxiv_id": "physics/0507192",
"authors": [
"Jeremy Butterfield"
],
"categories": [
"physics.class-ph",
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],
"title": "On Symmetry and Conserved Quantities in Classical Mechanics",
"url": "https://arxiv.org/abs/physics/0507192"
},
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