Symmetry

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.