On Symmetry and Conserved Quantities in Classical Mechanics
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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