The notion of continuous symmetry has largely and successfully been formalised in the mathematical notions of topological group, Lie group and group action. For most practical purposes, continuous symmetry is modelled by a group action of a topological group that preserves some structure. Particularly, let ${\displaystyle f:X\to Y}$  be a function, and G is a group that acts on X; then a subgroup ${\displaystyle H\subseteq G}$  is a symmetry of f if ${\displaystyle f(h\cdot x)=f(x)}$  for all ${\displaystyle h\in H}$ .

One-parameter subgroups edit

The simplest motions follow a one-parameter subgroup of a Lie group, such as the Euclidean group of three-dimensional space. For example translation parallel to the x-axis by u units, as u varies, is a one-parameter group of motions. Rotation around the z-axis is also a one-parameter group.

Continuous symmetry has a basic role in Noether's theorem in theoretical physics, in the derivation of conservation laws from symmetry principles, specifically for continuous symmetries. The search for continuous symmetries only intensified with the further developments of quantum field theory.

• Goldstone's theorem
• Infinitesimal transformation
• Noether's theorem
• Sophus Lie
• Motion (geometry)
• Circular symmetry

• Barker, William H.; Howe, Roger (2007). Continuous Symmetry: from Euclid to Klein. American Mathematical Society. ISBN 978-0-8218-3900-3.