Symmetry: Difference between revisions
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Symmetries are intimately related to [[conservation law]]s via [[Noether's theorem]]. | Symmetries are intimately related to [[conservation law]]s via [[Noether's theorem]]. | ||
[[Category: | [[Category:Transforms]] | ||
Revision as of 00:10, 15 August 2006
A symmetry of an equation is any operation which maps solutions to solutions; thus a symmetry is the same concept as a transform, except that the transformed equation is the same as the old.
In principle there are an infinite-dimensional space of symmetries; in practice, however, one works only with the finite-dimensional component of symmetries which have a clean and explicit algebraic description. Indeed many symmetries are linear in nature. Note that completely integrable equations enjoy an explicit infinite-dimensional space of symmetries, formed by using any of the infinite number of conserved quantities as a Hamiltonian.
The space of all symmetries form a group.
Symmetries are intimately related to conservation laws via Noether's theorem.
