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.

List of symmetries

Note that any given equation will typically only enjoy a subset of the symmetries on this list.

• Conformal symmetry
• Diffeomorphism symmetry
• Galilean symmetry
• Gauge invariance
• Lorentz symmetry
• Phase rotation symmetry
• Pseudoconformal symmetry
• Scaling symmetry
• Space rotation symmetry
• Space translation symmetry
• Time reversal symmetry
• Time translation symmetry