In

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries
Symmetry (from Ancient Greek, Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" ...

as motion
Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position
In physics, motion is the phenomenon in which an object changes its position (mathematics), position over time. Motion is mathematically described in terms of Displacem ...

s, as opposed to discrete symmetry, e.g. reflection symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflectio ...

, which is invariant under a kind of flip from one state to another. However, a discrete symmetry can always be reinterpreted as a subset of some higher-dimensional continuous symmetry, e.g. reflection of a 2 dimensional object in 3 dimensional space can be achieved by continuously rotating that object 180 degrees across a non-parallel plane.
Formalization

The notion of continuous symmetry has largely and successfully been formalised in the mathematical notions oftopological group
350px, The real numbers form a topological group under addition ">addition.html" ;"title="real numbers form a topological group under addition">real numbers form a topological group under addition
In mathematics, a topological group is a group ...

, Lie group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

and group action
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

. For most practical purposes continuous symmetry is modelled by a ''group action'' of a topological group that preserves some structure. Particularly, let $f:X\backslash to\; Y$ be a function, and ''G'' is a group that acts on ''X'' then a subgroup $H\backslash subseteq\; G$ is a symmetry of ''f'' if $f(h\backslash cdot\; x)\; =\; f(x)$ for all $h\backslash in\; H$.''
One-parameter subgroups

The simplest motions follow aone-parameter subgroup
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of a Lie group, such as the Euclidean group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of three-dimensional space
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...

. For example translation
Translation is the communication of the meaning
Meaning most commonly refers to:
* Meaning (linguistics), meaning which is communicated through the use of language
* Meaning (philosophy), definition, elements, and types of meaning discusse ...

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.
Noether's theorem

Continuous symmetry has a basic role inNoether's theorem
Noether's theorem or Noether's first theorem states that every differentiable
In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domai ...

in theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in contrast to experimental ph ...

, in the derivation of conservation law
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

s from symmetry principles, specifically for continuous symmetries. The search for continuous symmetries only intensified with the further developments of quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...

.
See also

*Goldstone's theorem
In particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, physical or chemical , chemica ...

*Infinitesimal transformation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

*Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable
In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domai ...

*Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norway, Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Biography
Marius Sop ...

*Motion (geometry)
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

*Circular symmetry
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

References

*William H. Barker, Roger Howe (2007), ''Continuous Symmetry: from Euclid to Klein'' {{DEFAULTSORT:Continuous Symmetry Symmetry Lie groups Group actions (mathematics)